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Writing numbers in written and expanded form
Quiz by Common Core - Grade 4
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What is an official invitation letter? The companies write a letter of invitation-business when they host business visitors from abroad or from the same region or country. The business visitors can be investors; potential buyers may be conference visitors, business partners, employees of any company, or mere individuals who come for training at the companyâs facilities. If a company is inviting any visitor, a representative of that company must write the letter. Also, the firms must have some specific people who would sign the invitation letters. These letters are very much precise, only containing the necessary information. The invitation letter should state the name of the business organization they represent and their relationship to the host (e.g., distributor, regional sales reps, etc.). The letter should articulate the planned dates of travel, and must be formatted professionally. What is a personal invitation letter? A Personal invitation letter is a letter one writes to invite people to a party or a social gathering at a very personal level. It is a formal request asking for the personâs presence at the event that is going to take place. All the relevant details regarding the event like the reason, date, time and venue and the dress code, if any, must be provided in the invitation letters. This will keep the guests informed, and they will feel happy to attend the event. The style and tone of the letter would depend upon the relationship between the sender and receiver. Through the letter, you should be able to make the receiver feel that you highly value his/her presence at the party or the event. A personal invitation letter can be written to invite a person to a birthday party, wedding, conference, meeting, dinner, etc. Before writing the letter, make sure you have a list of people whom you would like to invite to the party or the event. How to Write an Invitation Letter Writing an invitation letter becomes easy and swift once you get through the tips and the format of the invitation letter provided below. Usually block, semi-block or a modified block format is used for official invitation letters. The important aspects of any invitation letters are date, time, salutations and closing. For more advice refer to the tips provided. Tips for Invitation Letter Writing â Organize the Matter â Before you draft an invitation letter ensure that you have all the required material. This material refers to a list of the people to be invited, sequential order of the events, timings of the events, special guest, official documents, photocopies and any other required item. Some items may also need to be attached along with the letter, keep them alongside. Refer to these as and when required. All the relevant documents will help you in drafting the letter. â Drafting â You donât just write a letter straightway and post it. It has to be reviewed and finalized. One of these processes is drafting. Drafting ensures that your mistakes and their rectification arenât passed on to the invitation itself. Make all the mistakes in the draft itself. Drafting an invitation letter is important as sometimes we may make mistakes that we are not able to see but they are visible to others. One may require a draft to be approved by seniors before it is finalized. A second opinion from a friend or peer etc. may be required as well to determine certain things. â Politeness â You donât need to be told that you have to use polite language while writing an invitation letter, why would you be rude when sending an invitation? True, but you have to remind yourself of certain manners and etiquettes required of an invitation. Your invitation is your initiative, not the recipients so you need to be gracious. Always begin the letter with a welcome note instead of straightforward information of the invitation. Words of respect and gratitude are symbols of courtesy and politeness, always expressing your gratitude in the beginning and the end of the letter. â Positive Tone â The gesture of welcome and gratitude themselves are positive points of an invitation letter. Apart from these, gestures of appreciation and anticipation are other positive points which can persuade a guest to attend the event. When you show your appreciation and anticipation towards the recipient through your words, it is an acknowledgement of his importance and thereby a positive approach. Towards this effect two tenses are used within the invitation letter, the present and the future. The present tense conveys information about the event and the future tense conveys an anticipated presence of the guest. â Offer Assistance â An invitation being the responsibility of the sender, the assistance to the recipient by default becomes a responsibility of the host. The more facilities you provide the better the chances of someoneâs attendance. You can offer pick up and drop services, accommodation, meals, provide them contact numbers in case you are not present at the venue and other required assistance. Relevant facts like date, time and venue of the event in the beginning itself is itself assisting. These assistances encourage a positive response from the invitees. â Special Instructions â Some occasions require special instructions for the guests. These instructions can be: 1. Dress code 2. Road or route map 3. Purpose of the occasion â birthday, honor, anniversary etc. 4. Return gift 5. Response or confirmation to the invitation 6. Attire and items required for the guest to bring 7. No eatables allowed 8. Entrance only by invitation 9. 2 people per pass 10. No weapons allowed â Length of the Matter â A simple invitation letter will only contain only the relevant facts. A simple invitation letter features an introduction which allows the sender to introduce themselves and or the organization they represent. A simple background of the individual or company is enough. Though invitations are meant to be concise and straightforward, it isnât necessary. You can vary the length as per your need and requirement. Wedding and party invitation letters are not lengthy as compared to visit and certain personal invitation letters. â Using Letterhead â As a rule official Invitation letters require a letterhead. Letterhead represents the sender and its inclusion is authority established. If you have a pre printed letterhead then use that. Personal Invitation letters donât require letterheads and one can use it as per oneâs desire. â Gesture of Appreciation â Next, the appreciation for the guest to attend an activity or event must be shown. This can be completed with a formal note, stating that you look forward to seeing the individual at the event. â Donât forget the Enclosure â Some requests require certain documents to be attached; these can be the photocopies of documents like agreements, hard copies of email received, earlier correspondence, receipts, warranty etc. Keep original copies of all your letters, faxes, e-mails, and other related documents. â Closing the Letter â Start the letter with Gratitude and end it with the same. It is a professional and social courtesy. At the end of your last paragraph is written, a complimentary close of the likes of âSincerelyâ, âThank youâ, âTrulyâ is essential. Close the letter by restating your appreciation and gratitude. â Proofreading â Check for - awkward phrases, grammatical errors, incomplete sentences and spelling mistakes. Fix them with appropriate punctuation and remove dull or lifeless sentences and replace them with clever phrasing, poetry or a themed approach. This is the final step; the draft will be reviewed and revised before it acquires a proper form. Read it aloud to yourself to figure out mistakes which are missed out in writing. â Inform in Advance â Invitation letters need to be sent in advance. Try to send the invitation letter two weeks or more in advance. The recipient needs to know in advance so that they can adjust their schedules, book tickets or make other arrangements which are essential.
Broken windows are covered. Floorboards are patched and doors screwed back on. The road that was ruined by German tanks is shovelled and raked smooth. Boot-shaped bruises turn yellow then fade and disappear. Flowers grow and spread across the ugly German footprints stomped into garden beds. The village looks pretty once more. School stops for the summer and everyone is put to work on the kolkhoz, the village farm. Women and big boys begin harvesting the barley crops in the outer fields. The biggest girls milk the cows, morning and night, and keep the barns clean. Old Nikolay mends ploughs, horse harnesses, pitchforks and scythes in his workshop. Anna Pushinka teaches Yelena and her friends how to get the honey from the beehives that are scattered through the orchards. I am in charge of collecting eggs. My friends Olga and Nina help. Olga and Nina are five, a year younger than me. They are twins and look exactly alike, except Ninaâs nose is a little bit crooked from when she fell out of bed and squashed it sideways on the floor. The hens, ducks and geese wander free in the summer, so collecting eggs is like a treasure hunt and takes hours. Catching the hens for their daily hugs takes even longer, but I think itâs important because hugs make everyone happy and happy hens lay bigger eggs. Olga says Iâm the best hen-hugger in all of Russia. Nina says Iâll be the best cow-hugger, too, when my arms grow longer. But good hugs have nothing to do with the size of your arms. Itâs all to do with the size of your heart. When we are done with the hens, Olga, Nina and I can spend the rest of the day doing whatever we like. We climb the apricot trees, chase squirrels, lie in the meadow marvelling at how hot Ushankaâs black fur becomes in the sunshine, make daisy chains and race little boats of bark in the stream. I teach Olga and Nina the alphabet and we use charcoal to write our letters and our names all over the village â on doors and walls and the freshly cut ends of firewood. In between, I practise my knots. In case the German princemonsters return. I slip into Old Nikolayâs workshop and tie knots in the harnesses hanging on the walls. I wander into gardens where the washing is hung out to dry and tie knots in the laces on pants and smocks. I creep up behind Anna Pushinka and tie knots in her apron strings. I find baling twine in the hay shed and tie my own ankles together. I do such a good job of these last knots that I canât get them undone. I have to jump all the way to Olga and Ninaâs house and ask them to cut me free with their mamaâs knife. At the end of each day, Ushanka and I run out into the distant barley fields to meet Mama. This is my favourite part of the day, because Mama always shouts, âLittle Rabbit!â and smothers my head with kisses. And as we walk home, we sing. Everyone â women, big boys and me. I love to sing. Almost as much as I love to be kissed by Mama. Sometimes one of the boys, Mikhail, has his balalaika with him. He takes the instrument out from beneath the sheaves of barley piled high on the wagon and plays music. We sing about forests and orchards and people who find their true love. As we walk home, arm in arm, my heart fills with happiness and my belly swells with pride that I am allowed to sing along with the big boys. And I can almost forget about the German prince-monsters and their lies about Russia and their big ugly boots. Almost. But today, when Mikhail reaches for his balalaika, I see other things hiding beneath the barley sheaves. Three of the mamas rush forward and cover them up, but itâs too late. I know they are there. Iâve already seen them. Rifles. Lots of rifles. Mikhail hugs his balalaika to his chest and blushes. âSo play!â cries Mama, her voice oddly loud and high. âLetâs play Sashaâs favourite song, âThe Little Birch Treeâ.â So Mikhail plays and everyone sings about the lovely birch tree with its curly leaves and the branches that will be turned into silver flutes. They sing too quickly, too loudly, and as they sing and walk, they cast nervous sideways glances at me. âItâs alright,â I say, when the song comes to an end. âI didnât see the rifles.â Mama nods and smiles, and I know it was the right thing to say. But I did see the rifles. And I think about Yelena wanting to get lots of guns and dynamite for the Partisans so they can shoot the Germans and blow them into thousands of tiny pieces, and Mama looking as though she agreed, and I know this is what the mamas and the big boys are doing. As well as harvesting, they are helping the Partisans. Three days later, I wake before dawn and I am all alone. Yelena is always here beside me when I wake. But not this morning. I climb down from our bed above the stove. Mama is filling a cloth sack with bread. She ties it closed with a piece of string and hands it to Yelena. âStay out of sight,â says Mama. âAnd donât return until after dark.â âWhereâs she going?â I ask. âNowhere,â snaps Mama. âThen why does she need all that bread?â I ask. âThereâs nothing left for us.â Mama baked four loaves last night and she has stuffed them all into the sack. Yelena opens her mouth, but before she can speak, Mama shoves her out the door and sends her on the way to nowhere. Mama turns and stares at me, her blue, blue cornflower eyes wide with worry. âI know,â I say, flopping down on the bench. âI didnât see any bread.â Mama sits beside me and takes my hand. âAnd . . .?â she prods, obviously waiting for more. I puzzle for a while, then say, âAnd I donât have a sister called Yelena.â Mama laughs, softly and with a little bit of sadness around the edges. âSweet Little Rabbit! You do have a sister called Yelena.â âI do?â I ask, now confused. âI havenât seen the rifles or the bread, but I have seen Yelena?â âYes.â Mama smiles and the magic makes me smile, too. And I am glad that Yelena is real because I love her very much. âYelena is real,â Mama explains, âbut she does not carry sacks of bread into the forest for the Partisans.â âOf course not!â I shout, slapping my forehead. âBecause there is no bread!â Mama laughs loudly now, with not a hint of sadness. She hugs me, pressing me against her warm, loving heart, covering my head with kisses. âClever Little Rabbit,â she murmurs, and then, in barely a whisper, âYour papa would be so proud.â When I wake the next morning, Yelena is sleeping beside me, her mouth open, her braided hair unravelling. Mama is serving kasha to a strange woman seated at our table. I crawl down from above the stove and slide along the bench beside her. I stare at her pants, her tunic, the rope she is using as a belt and her big boots. Sheâs dressed like a man! And thereâs a rifle leaning against the wall near the door. âHello,â I say. âIâm Sasha.â The woman doesnât reply. She just shovels down her kasha. I line my four wooden bears along the table in front of her bowl and say, âThese are my bears: Big Bear, Medium Bear, Little Bear and Even Littler Bear.â âHello, Sasha. Hello, bears.â She smiles but she doesnât tell me her name. âWhy are you dressed like a man?â I ask, tugging at the sleeve of her tunic. âBecause menâs clothes make it easier to run and climb and crawl and shoot,â she says. âYouâre a Partisan!â I gasp. âBut sheâs not real,â says Mama, placing a bowl of kasha before me. âIs the kasha real?â I ask. Mama laughs. âYes, Little Rabbit.â Iâm glad the food is real, because Iâm hungry. But Iâm disappointed that the woman is not real. I was going to ask if I could use her rope-belt to tie her ankles together. For practice. But if sheâs not real, then the rope and her ankles arenât either. The woman finishes her kasha, hangs her rifle over her shoulder, kisses Mama on the cheek then slips out the door. I run to the window to watch her leave, but by the time I get there, sheâs gone. Vanished. âBecause sheâs not real,â I whisper. A week later, Mama and I are working in the garden. We sing as we weed between the flowers and pluck caterpillars from the vegetables. Anna Pushinka is picking strawberries in her garden and wanders over. âTaste these,â she says, holding out the basket. Mama reaches in and takes out a fat strawberry and a tiny piece of folded paper. The strawberry goes into her mouth, the paper into her pocket. âWhatâs on the paper?â I ask. âPaper?â Anna Pushinka replies with a wave of her hand. âGoodness, Sasha! Who has money for paper? These are lean times. We must choose between paper for writing and noodles for our soup. And I always choose noodles.â She chuckles and I know the paper is yet another thing that is not real. That night, Mama slips the paper to Yelena, but she drops it on the floor. I pick it up for her, and I see that there are tiny words and numbers written all over it. I wish I could read better. Iâm desperate to know what it says. Or rather, what it doesnât say, because itâs not real. Later, when Mama has tucked us into our bed above the stove and Ushanka has wrapped herself around the top of my head, I ask Yelena, âWhatâs on the paper?â âWhat paper?â says Yelena. âThe paper that isnât real,â I reply. Yelena stares at me, nibbling her lip, then whispers, âA message for the Partisans. Stuff about where the Germans have their headquarters and when their trains are travelling and where they store their ammunition.â âWhy?â âSo the Partisans can blow them up.â Yelena grabs my arm. âBut donât tell anyone. Itâs a secret.â âWhatâs a secret?â I ask. âThe message.â âWhat message?â I say, my eyes wide. Yelena laughs. âGood boy, Sasha.â My belly swells with pride. I know how to play this game. âHow are your knots coming along?â asks Yelena. âGood! Yesterday, I crept into the dairy and tied knots in the apron strings of all the girls who were milking and only one of them noticed. Today, I tied Olgaâs ankles together with Mamaâs embroidery thread and just now, while you were taking a bath, I tied the sleeves of your blouse together in an enormous knot.â Yelena rolls her eyes, then says, âIâll see if I can find you some rope for practising.â âPractising what?â I ask. âYour knots,â she says. âWhat knots?â Yelena, my big sister who is twelve and always serious t
Alright, Isti â hereâs a longer and more detailed English version of the Isaac Newton text, still written at a level thatâs accessible for Grade 4 students, but rich enough in information to meet PISA literacy expectations and EF A2-level vocabulary. Iâve kept sentences short, clear, and with explanations for new concepts so itâs easier for young learners to follow, while still including both famous facts and lesser-known stories. ⸝ Isaac Newton: The Man Who Changed the Way We See the World A Boy from a Small Village Isaac Newton was born on January 4, 1643, in Woolsthorpe, a small village in England. His life was not easy. His father died before he was born. When he was just a few months old, his mother remarried and left him to live with his grandmother. Isaac missed his parents, but he kept himself busy by making things and exploring the world around him. As a child, Isaac liked to build models and machines. He made a small windmill that could turn with the wind. He built a water clock that told the time by dripping water into a container. He even made a sundial â a clock that tells the time by using the shadow of the sun. đĄ Did you know? The sundial marks that Isaac carved as a boy can still be seen today on the wall of his old house. ⸝ School and Curiosity When Newton first went to school, he was not the top student. At first, he did not pay much attention in class. But one day, another boy teased him for not being smart. Newton decided to study hard to prove him wrong. Soon, he became the best in his class. Isaac loved asking questions. He wanted to know how and why things happened. He enjoyed watching the stars at night and thinking about how the world worked. ⸝ The Falling Apple and Gravity One of the most famous stories about Newton is the falling apple. One afternoon, Isaac sat in his motherâs garden and saw an apple drop from a tree. This made him think: âWhy does the apple fall straight down? Why doesnât it fly up into the sky?â From this question, Newton began to think about gravity â an invisible force that pulls objects toward each other. Gravity is what keeps our feet on the ground. Itâs also what keeps the Moon moving around the Earth and the planets moving around the Sun. đĄ Fun fact: The apple did not hit Newtonâs head. Thatâs just a story people made up later to make the tale more exciting. ⸝ Newtonâs Three Laws of Motion Newton studied movement and wrote three important rules: 1. Objects stay still or keep moving unless something makes them change. ⢠Example: A ball will not roll unless you push it. 2. The bigger the push, the bigger the movement. ⢠Example: If you kick a ball harder, it will go faster and farther. 3. Every action has an equal and opposite reaction. ⢠Example: When you jump off a boat, the boat moves backward as you move forward. These three laws are still used today to understand how cars, rockets, and even roller coasters work. ⸝ Discoveries in Light and Color Newton also studied light. He found that white light is not just one color â it is made of many colors. He used a glass prism to split sunlight into a rainbow. This helped scientists understand how colors work. ⸝ Inventions and New Ideas Newton made a special telescope that used mirrors instead of lenses. This type of telescope made images of planets and stars much clearer. It is still called the Newtonian telescope today. He also worked in mathematics and helped create a new type of math called calculus, which is used to study changes and movement. ⸝ Strange Experiments Newton was so curious that he sometimes tested ideas on himself. Once, he put a thin needle, called a bodkin, beside his eye to see how it would change his vision. It was very dangerous, but luckily he did not go blind. đĄ Did you know? Newton also studied alchemy â an old kind of science where people tried to turn metal into gold. He never succeeded, but it showed how wide his interests were. ⸝ Later Life and Work At the age of 27, Newton became a professor at Cambridge University. He later worked for the Royal Mint, making sure coins were made safely and stopping people from making fake money. He was very strict, and some criminals were sent to prison because of his work. Newton never married. He spent most of his life reading, writing, and doing experiments. ⸝ The End of His Life Isaac Newton died in 1727 at the age of 84. He was buried in Westminster Abbey, a famous place in London where great people of Britain are honored. His work changed the world forever. Even today, scientists, engineers, and students still use Newtonâs laws and ideas. đŹ Newton once said: âIf I have seen further, it is by standing on the shoulders of giants.â This means we can make new discoveries by learning from the work of others who came before us. give 10 questions to each passage with PISA literacy standard for kid 10 years, 1. Nikola Tesla: The Man Who Dreamed of Lightning Born: July 10, 1856 Died: January 7, 1943 When Nikola Tesla was a boy in Croatia, he saw a flash of lightning and asked his mother, âCan we catch the light?â That question never left him. As he grew older, Tesla became a brilliant inventor, especially fascinated by electricity. He believed in a future where energy could be sent wirelessly through the airâlike music through the radio! Tesla invented the alternating current (AC) system, which became the foundation of modern electricity. At the time, Thomas Edison promoted direct current (DC), and the two men had a fierce competition. Many laughed at Tesla's bold ideas, but he never gave up. He dreamed of wireless communication, flying machines, and even free energy for everyone. Though he died alone and poor, today the world honors his vision. Think About It: Why do you think people didnât believe Tesla at first? What can we learn from Teslaâs courage to dream big? 2. Charles Darwin: The Man Who Studied the Worldâs Weirdest Creatures Born: February 12, 1809 Died: April 19, 1882 When young Charles Darwin got on a ship called HMS Beagle, he didnât know he would change science forever. He sailed around the world for five years, collecting plants, animals, and fossils. On the GalĂĄpagos Islands, he noticed something curious: finches had different beaks depending on their island. Why? Darwinâs observations led him to write the theory of evolution by natural selection. It explained how animals adapt and survive. But his ideas shocked many people because they seemed to challenge religious beliefs. Despite the controversy, Darwin continued his work. His book On the Origin of Species changed how we see life on Earth. Think About It: Should scientists share their ideas even if they go against what others believe? How did traveling help Darwin make new discoveries? 3. Marie Curie: The Woman Who Glowed in the Dark Born: November 7, 1867 Died: July 4, 1934 Marie Curie was born in Poland at a time when girls were not allowed to study science. But that didnât stop her. She moved to France, worked day and night, and discovered radioactivity, a powerful energy hidden inside atoms. She and her husband, Pierre Curie, found two new elements: polonium and radium. She became the first woman to win a Nobel Prize, and the only person to win in two different sciences: physics and chemistry. Even when Pierre died in an accident, Marie continued their work. Her discoveries helped doctors treat cancerâbut working with radioactive materials also harmed her health. She died from radiation exposure, but her legacy lives on. Think About It: What challenges did Marie Curie face as a woman in science? Why is it important to balance discovery with safety? 4. Galileo Galilei: The Star Watcher Who Defied the Church Born: February 15, 1564 Died: January 8, 1642 Galileo loved looking at the stars. He built one of the first powerful telescopes and made stunning discoveries: mountains on the Moon, moons around Jupiter, and that the Earth orbits the Sunânot the other way around. This idea, called heliocentrism, went against the teachings of the Church. He was put on trial and forced to say he was wrong. But he wasnât. He spent his last years under house arrest, quietly writing. Today, Galileo is called the father of modern science for daring to question what others blindly believed. Think About It: Why do you think Galileo was punished for telling the truth? Should science always follow evidence, even if it goes against powerful beliefs? 5. Isaac Newton: The Man Who Asked âWhy?â When an Apple Fell Born: January 4, 1643 Died: March 31, 1727 One day, an apple fell from a tree, and Isaac Newton began to wonder: Why did it fall down, not sideways or up? This simple question led to his theory of gravity. Newton also invented calculus, described the laws of motion, and changed physics forever. But Newton wasnât just a geniusâhe was curious, quiet, and often worked alone. He believed everything in nature followed rules, and it was our job to discover them. Thanks to him, we understand how planets move, how rockets launch, and why you fall when you trip. Think About It: How did Newtonâs curiosity lead to great discoveries? Do you think working alone helped or hurt Newton? 6. Ada Lovelace: The First Computer Programmer Before Computers Existed Born: December 10, 1815 Died: November 27, 1852 Ada Lovelace was the daughter of the famous poet Lord Byron, but she didnât love poetryâshe loved numbers! At a time when girls were expected to sew, Ada studied mathematics. She met Charles Babbage, who designed an early computer called the Analytical Engine. Ada imagined the machine could do more than just mathâit could create music, art, and even write! She wrote what is now considered the first computer program, long before real computers were built. Think About It: How did Ada imagine something that didnât exist yet? Why do we call her a pioneer in technology? 7. Albert Einstein: The Man Who Brought Time and Space Together Born: March 14, 1879 Died: April 18, 1955 Albert Einstein wasnât always a good student. In fact, his teachers thought he was slow. But Einstein thought deeply. He asked big questions like, âWhat if you could ride a beam of light?â His theories of relativity changed how we see space, time, and gravity. He also warned the world about the dangers of nuclear weapons, even though his ideas helped create them. Einstein believed science should help people, not harm them. With his messy hair, kind smile, and brilliant mind, he remains a symbol of genius. Think About It: Can someone be bad in school but still be brilliant? Should scientists be responsible for how their inventions are used? 8. Pythagoras: The Musician Who Loved Math Born: Around 570 BC Died: Around 495 BC Long ago in ancient Greece, Pythagoras believed the universe followed numbers. He discovered the Pythagorean Theorem, a rule about triangles that helps us build houses, design computers, and navigate space. He also believed that music had math inside itâthat certain notes made perfect harmony because of mathematical ratios. Pythagoras started a secret school and taught his students to search for truth through numbers, shapes, and sound. Think About It: Why do you think Pythagoras saw math in everything? How does music relate to math? 9. Rosalind Franklin: The Woman Behind the DNA Discovery Born: July 25, 1920 Died: April 16, 1958 Rosalind Franklin loved looking closely at things. She used a special machine called X-ray crystallography to photograph molecules. One of her greatest photos, called Photo 51, showed the shape of DNA, the molecule that carries lifeâs instructions. But her work was taken without credit. Two men, Watson and Crick, used her photo to build their famous model of DNA and won the Nobel Prize. Rosalind died young and never knew how important her work became. Think About It: Why is it important to give credit in science? What can we learn from Rosalindâs quiet strength? 10. Carl Linnaeus: The Man Who Gave Names to Everything Born: May 23, 1707 Died: January 10, 1778 Have you ever wondered why a tiger is called Panthera tigris? Thatâs thanks to Carl Linnaeus, a Swedish scientist who created a way to name and organize every living thing. His system is still used today in biology. Linnaeus loved nature and spent his life collecting plants, animals, and even rocks. He believed that by organizing life, we could better understand it. Thanks to him, we now have a global âdictionary of nature.â Think About It: Why is it important to name and organize living things? How does order help us understand the world?
Covalent Molecules and Compounds Just as an atom is the simplest unit that has the fundamental chemical properties of an element, a molecule is the simplest unit that has the fundamental chemical properties of a covalent compound. Some pure elements exist as covalent molecules. Hydrogen, nitrogen, oxygen, and the halogens occur naturally as the diatomic (âtwo atomsâ) molecules H2, N2, O2, F2, Cl2, Br2, and I2 (part (a) in Figure 3.1.1). Similarly, a few pure elements exist as polyatomic (âmany atomsâ) molecules, such as elemental phosphorus and sulfur, which occur as P4 and S8 (part (b) in Figure 3.1.1). Each covalent compound is represented by a molecular formula, which gives the atomic symbol for each component element, in a prescribed order, accompanied by a subscript indicating the number of atoms of that element in the molecule. The subscript is written only if the number of atoms is greater than 1. For example, water, with two hydrogen atoms and one oxygen atom per molecule, is written as H2O. Similarly, carbon dioxide, which contains one carbon atom and two oxygen atoms in each molecule, is written as CO2. Covalent compounds that predominantly contain carbon and hydrogen are called organic compounds. The convention for representing the formulas of organic compounds is to write carbon first, followed by hydrogen and then any other elements in alphabetical order (e.g., CH4O is methyl alcohol, a fuel). Compounds that consist primarily of elements other than carbon and hydrogen are called inorganic compounds; they include both covalent and ionic compounds. In inorganic compounds, the component elements are listed beginning with the one farthest to the left in the periodic table, as in CO2 or SF6. Those in the same group are listed beginning with the lower element and working up, as in ClF. By convention, however, when an inorganic compound contains both hydrogen and an element from groups 13â15, hydrogen is usually listed last in the formula. Examples are ammonia (NH3) and silane (SiH4). Compounds such as water, whose compositions were established long before this convention was adopted, are always written with hydrogen first: Water is always written as H2O, not OH2. The conventions for inorganic acids, such as hydrochloric acid (HCl) and sulfuric acid (H2SO4), are described elswhere. Note! For organic compounds: write C first, then H, and then the other elements in alphabetical order. For molecular inorganic compounds: start with the element at far left in the periodic table; list elements in same group beginning with the lower element and working up. Write the molecular formula of each compound. a. The phosphorus-sulfur compound that is responsible for the ignition of so-called strike anywhere matches has 4 phosphorus atoms and 3 sulfur atoms per molecule. b. Ethyl alcohol, the alcohol of alcoholic beverages, has 1 oxygen atom, 2 carbon atoms, and 6 hydrogen atoms per molecule. c. Freon-11, once widely used in automobile air conditioners and implicated in damage to the ozone layer, has 1 carbon atom, 3 chlorine atoms, and 1 fluorine atom per molecule. Solution: a. ⢠A The molecule has 4 phosphorus atoms and 3 sulfur atoms. Because the compound does not contain mostly carbon and hydrogen, it is inorganic. ⢠B Phosphorus is in group 15, and sulfur is in group 16. Because phosphorus is to the left of sulfur, it is written first. ⢠C Writing the number of each kind of atom as a right-hand subscript gives P4S3 as the molecular formula. b. ⢠A Ethyl alcohol contains predominantly carbon and hydrogen, so it is an organic compound. ⢠B The formula for an organic compound is written with the number of carbon atoms first, the number of hydrogen atoms next, and the other atoms in alphabetical order: CHO. ⢠C Adding subscripts gives the molecular formula C2H6O. c. ⢠A Freon-11 contains carbon, chlorine, and fluorine. It can be viewed as either an inorganic compound or an organic compound (in which fluorine has replaced hydrogen). The formula for Freon-11 can therefore be written using either of the two conventions. ⢠B According to the convention for inorganic compounds, carbon is written first because it is farther left in the periodic table. Fluorine and chlorine are in the same group, so they are listed beginning with the lower element and working up: CClF. Adding subscripts gives the molecular formula CCl3F. ⢠C We obtain the same formula for Freon-11 using the convention for organic compounds. The number of carbon atoms is written first, followed by the number of hydrogen atoms (zero) and then the other elements in alphabetical order, also giving CCl3F. Write the molecular formula for each compound. a. Nitrous oxide, also called âlaughing gas,â has 2 nitrogen atoms and 1 oxygen atom per molecule. Nitrous oxide is used as a mild anesthetic for minor surgery and as the propellant in cans of whipped cream. b. Sucrose, also known as cane sugar, has 12 carbon atoms, 11 oxygen atoms, and 22 hydrogen atoms. c. Sulfur hexafluoride, a gas used to pressurize âunpressurizedâ tennis balls and as a coolant in nuclear reactors, has 6 fluorine atoms and 1 sulfur atom per molecule. Answer: a. N2O b. C12H22O11 c. SF6. Ionic Compounds The substances described in the preceding discussion are composed of molecules that are electrically neutral; that is, the number of positively-charged protons in the nucleus is equal to the number of negatively-charged electrons. In contrast, ions are atoms or assemblies of atoms that have a net electrical charge. Ions that contain fewer electrons than protons have a net positive charge and are called cations. Conversely, ions that contain more electrons than protons have a net negative charge and are called anions. Ionic compounds contain both cations and anions in a ratio that results in no net electrical charge. Note! Ionic compounds contain both cations and anions in a ratio that results in zero electrical charge.An ionic compound that contains only two elements, one present as a cation and one as an anion, is called a binary ionic compound. One example is MgCl2, a coagulant used in the preparation of tofu from soybeans. For binary ionic compounds, the subscripts in the empirical formula can also be obtained by crossing charges: use the absolute value of the charge on one ion as the subscript for the other ion. This method is shown schematically as follows: Crossing charges. One method for obtaining subscripts in the empirical formula is by crossing charges. When crossing charges, it is sometimes necessary to reduce the subscripts to their simplest ratio to write the empirical formula. Consider, for example, the compound formed by Mg2+ and O2â. Using the absolute values of the charges on the ions as subscripts gives the formula Mg2O2:Polyatomic Ions Polyatomic ions are groups of atoms that bear net electrical charges, although the atoms in a polyatomic ion are held together by the same covalent bonds that hold atoms together in molecules. Just as there are many more kinds of molecules than simple elements, there are many more kinds of polyatomic ions than monatomic ions. Two examples of polyatomic cations are the ammonium (NH4+) and the methylammonium (CH3NH3+) ions. P. The method used to predict the empirical formulas for ionic compounds that contain monatomic ions can also be used for compounds that contain polyatomic ions. The overall charge on the cations must balance the overall charge on the anions in the formula unit. Thus, K+ and NO3â ions combine in a 1:1 ratio to form KNO3 (potassium nitrate or saltpeter), a major ingredient in black gunpowder. Similarly, Ca2+ and SO42â form CaSO4 (calcium sulfate), which combines with varying amounts of water to form gypsum and plaster of Paris. The polyatomic ions NH4+ and NO3â form NH4NO3 (ammonium nitrate), a widely used fertilizer and, in the wrong hands, an explosive. One example of a compound in which the ions have charges of different magnitudes is calcium phosphate, which is composed of Ca2+ and PO43â ions; it is a major component of bones. The compound is electrically neutral because the ions combine in a ratio of three Ca2+ ions [3(+2) = +6] for every two ions [2(â3) = â6], giving an empirical formula of Ca3(PO4)2; the parentheses around PO4 in the empirical formula indicate that it is a polyatomic ion. Writing the formula for calcium phosphate as Ca3P2O8 gives the correct number of each atom in the formula unit, but it obscures the fact that the compound contains readily identifiable PO43â ions.Summary ⢠There are two fundamentally different kinds of chemical bonds (covalent and ionic) that cause substances to have very different properties. ⢠The composition of a compound is represented by an empirical or molecular formula, each consisting of at least one formula unit.Contributors The atoms in chemical compounds are held together by attractive electrostatic interactions known as chemical bonds. Ionic compounds contain positively and negatively charged ions in a ratio that results in an overall charge of zero. The ions are held together in a regular spatial arrangement by electrostatic forces. Most covalent compounds consist of molecules, groups of atoms in which one or more pairs of electrons are shared by at least two atoms to form a covalent bond. The atoms in molecules are held together by the electrostatic attraction between the positively charged nuclei of the bonded atoms and the negatively charged electrons shared by the nuclei. The molecular formula of a covalent compound gives the types and numbers of atoms present. Compounds that contain predominantly carbon and hydrogen are called organic compounds, whereas compounds that consist primarily of elements other than carbon and hydrogen are inorganic compounds. Diatomic molecules contain two atoms, and polyatomic molecules contain more than two. A structural formula indicates the composition and approximate structure and shape of a molecule. Single bonds, double bonds, and triple bonds are covalent bonds in which one, two, and three pairs of electrons, respectively, are shared between two bonded atoms. Atoms or groups of atoms that possess a net electrical charge are called ions; they can have either a positive charge (cations) or a negative charge (anions). Ions can consist of one atom (monatomic ions) or several (polyatomic ions). The charges on monatomic ions of most main group elements can be predicted from the location of the element in the periodic table. Ionic compounds usually form hard crystalline solids with high melting points. Covalent molecular compounds, in contrast, consist of discrete molecules held together by weak intermolecular forces and can be gases, liquids, or solids at room temperature and pressure. An empirical formula gives the relative numbers of atoms of the elements in a compound, reduced to the lowest whole numbers. The formula unit is the absolute grouping represented by the empirical formula of a compound, either ionic or covalent. Empirical formulas are particularly useful for describing the composition of ionic compounds, which do not contain readily identifiable molecules. Some ionic compounds occur as hydrates, which contain specific ratios of loosely bound water molecules called waters of hydration.
Introduction to Free Fall A free-falling object is an object that is falling under the sole influence of gravity. Any object that is being acted upon only by the force of gravity is said to be in a state of free fall. There are two important motion characteristics that are true of free-falling objects: ⢠Free-falling objects do not encounter air resistance. ⢠All free-falling objects (on Earth) accelerate downwards at a rate of 9.8 m/s/s (often approximated as 10 m/s/s for back-of-the-envelope calculations) Because free-falling objects are accelerating downwards at a rate of 9.8 m/s/s, a ticker tape trace or dot diagram of its motion would depict an acceleration. The dot diagram at the right depicts the acceleration of a free-falling object. The position of the object at regular time intervals - say, every 0.1 second - is shown. The fact that the distance that the object travels every interval of time is increasing is a sure sign that the ball is speeding up as it falls downward. Recall from an earlier lesson, that if an object travels downward and speeds up, then its acceleration is downward. Free-fall acceleration is often witnessed in a physics classroom by means of an ever-popular strobe light demonstration. The room is darkened and a jug full of water is connected by a tube to a medicine dropper. The dropper drips water and the strobe illuminate the falling droplets at a regular rate - say once every 0.2 seconds. Instead of seeing a stream of water free-falling from the medicine dropper, several consecutive drops with increasing separation distance are seen. The pattern of drops resembles the dot diagram shown in the graphic at the right. The Acceleration of Gravity It was learned in the previous part of this lesson that a free-falling object is an object that is falling under the sole influence of gravity. A free-falling object has an acceleration of 9.8 m/s/s, downward (on Earth). This numerical value for the acceleration of a free-falling object is such an important value that it is given a special name. It is known as the acceleration of gravity - the acceleration for any object moving under the sole influence of gravity. A matter of fact, this quantity known as the acceleration of gravity is such an important quantity that physicists have a special symbol to denote it - the symbol g. The numerical value for the acceleration of gravity is most accurately known as 9.8 m/s2. There are slight variations in this numerical value (to the second decimal place) that are dependent primarily upon on altitude. We will occasionally use the approximated value of 10 m/s2 in order to reduce the complexity of the many mathematical tasks that we will perform with this number. By so doing, we will be able to better focus on the conceptual nature of physics without too much of a sacrifice in numerical accuracy. g = 9.8 m/s2, downward Look It Up! Even on the surface of the Earth, there are local variations in the value of the acceleration of gravity (g). These variations are due to latitude, altitude and the local geological structure of the region. Recall from an earlier lesson that acceleration is the rate at which an object changes its velocity. It is the ratio of velocity change to time between any two points in an object's path. To accelerate at 9.8 m/s2 means to change the velocity by 9.8 m/s each second. If the velocity and time for a free-falling object being dropped from a position of rest were tabulated, then one would note the following pattern. Time (s) Velocity (m/s) 0 0 1 - 9.8 2 - 19.6 3 - 29.4 4 - 39.2 5 - 49.0 . Observe that the velocity-time data above reveal that the object's velocity is changing by 9.8 m/s each consecutive second. That is, the free-falling object has an acceleration of approximately 9.8 m/s2. Another way to represent this acceleration of 9.8 m/s2 is to add numbers to our dot diagram that we saw earlier in this lesson. The velocity of the ball is seen to increase as depicted in the diagram at the right. (NOTE: The diagram is not drawn to scale - in two seconds, the object would drop considerably further than the distance from shoulder to toes.) Representing Free Fall by Graphs ⢠Early in Lesson 1 it was mentioned that there are a variety of means of describing the motion of objects. One such means of describing the motion of objects is through the use of graphs - position versus time and velocity vs. time graphs. In this part of Lesson 5, the motion of a free-falling motion will be represented using these two basic types of graphs. Representing Free Fall by Position-Time Graphs A position versus time graph for a free-falling object is shown below. Observe that the line on the graph curves. As learned earlier, a curved line on a position versus time graph signifies an accelerated motion. Since a free-falling object is undergoing an acceleration (g = 9.8 m/s/s), it would be expected that its position-time graph would be curved. A further look at the position-time graph reveals that the object starts with a small velocity (slow) and finishes with a large velocity (fast). Since the slope of any position vs. time graph is the velocity of the object (as learned in Lesson 3), the small initial slope indicates a small initial velocity and the large final slope indicates a large final velocity. Finally, the negative slope of the line indicates a negative (i.e., downward) velocity. Representing Free Fall by Velocity-Time Graphs A velocity versus time graph for a free-falling object is shown below. Observe that the line on the graph is a straight, diagonal line. As learned earlier, a diagonal line on a velocity versus time graph signifies an accelerated motion. Since a free-falling object is undergoing an acceleration (g = 9,8 m/s/s, downward), it would be expected that its velocity-time graph would be diagonal. A further look at the velocity-time graph reveals that the object starts with a zero velocity (as read from the graph) and finishes with a large, negative velocity; that is, the object is moving in the negative direction and speeding up. An object that is moving in the negative direction and speeding up is said to have a negative acceleration (if necessary, review the vector nature of acceleration). Since the slope of any velocity versus time graph is the acceleration of the object (as learned in Lesson 4), the constant, negative slope indicates a constant, negative acceleration. This analysis of the slope on the graph is consistent with the motion of a free-falling object - an object moving with a constant acceleration of 9.8 m/s/s in the downward direction. The Kinematic Equations The goal of this first unit has been to investigate the variety of means by which the motion of objects can be described. The variety of representations that we have investigated includes verbal representations, pictorial representations, numerical representations, and graphical representations (position-time graphs and velocity-time graphs). In Lesson 6, we will investigate the use of equations to describe and represent the motion of objects. These equations are known as kinematic equations. There are a variety of quantities associated with the motion of objects - displacement (and distance), velocity (and speed), acceleration, and time. Knowledge of each of these quantities provides descriptive information about an object's motion. For example, if a car is known to move with a constant velocity of 22.0 m/s, North for 12.0 seconds for a northward displacement of 264 meters, then the motion of the car is fully described. And if a second car is known to accelerate from a rest position with an eastward acceleration of 3.0 m/s2 for a time of 8.0 seconds, providing a final velocity of 24 m/s, East and an eastward displacement of 96 meters, then the motion of this car is fully described. These two statements provide a complete description of the motion of an object. However, such completeness is not always known. It is often the case that only a few parameters of an object's motion are known, while the rest are unknown. For example as you approach the stoplight, you might know that your car has a velocity of 22 m/s, East and is capable of a skidding acceleration of 8.0 m/s2, West. However you do not know the displacement that your car would experience if you were to slam on your brakes and skid to a stop; and you do not know the time required to skid to a stop. In such an instance as this, the unknown parameters can be determined using physics principles and mathematical equations (the kinematic equations). The BIG 4 The kinematic equations are a set of four equations that can be utilized to predict unknown information about an object's motion if other information is known. The equations can be utilized for any motion that can be described as being either a constant velocity motion (an acceleration of 0 m/s/s) or a constant acceleration motion. They can never be used over any time period during which the acceleration is changing. Each of the kinematic equations include four variables. If the values of three of the four variables are known, then the value of the fourth variable can be calculated. In this manner, the kinematic equations provide a useful means of predicting information about an object's motion if other information is known. For example, if the acceleration value and the initial and final velocity values of a skidding car is known, then the displacement of the car and the time can be predicted using the kinematic equations. Lesson 6 of this unit will focus upon the use of the kinematic equations to predict the numerical values of unknown quantities for an object's motion. The four kinematic equations that describe an object's motion are: There are a variety of symbols used in the above equations. Each symbol has its own specific meaning. The symbol d stands for the displacement of the object. The symbol t stands for the time for which the object moved. The symbol a stands for the acceleration of the object. And the symbol v stands for the velocity of the object; a subscript of i after the v (as in vi) indicates that the velocity value is the initial velocity value and a subscript of f (as in vf) indicates that the velocity value is the final velocity value. Each of these four equations appropriately describes the mathematical relationship between the parameters of an object's motion. As such, they can be used to predict unknown information about an object's motion if other information is known. In the next part of Lesson 6 we will investigate the process of doing this. Kinematic Equations and Problem-Solving The four kinematic equations that describe the mathematical relationship between the parameters that describe an object's motion were introduced in the previous part of Lesson 6. The four kinematic equations are: In the above equations, the symbol d stands for the displacement of the object. The symbol t stands for the time for which the object moved. The symbol a stand for the acceleration of the object. And the symbol v stands for the instantaneous velocity of the object; a subscript of i after the v (as in vi) indicates that the velocity value is the initial velocity value and a subscript of f (as in vf) indicates that the velocity value is the final velocity value. Problem-Solving Strategy In this part of Lesson 6 we will investigate the process of using the equations to determine unknown information about an object's motion. The process involves the use of a problem-solving strategy that will be used throughout the course. The strategy involves the following steps: 1. Construct an informative diagram of the physical situation. 2. Identify and list the given information in variable form. 3. Identify and list the unknown information in variable form. 4. Identify and list the equation that will be used to determine unknown information from known information. 5. Substitute known values into the equation and use appropriate algebraic steps to solve for the unknown information. 6. Check your answer to ensure that it is reasonable and mathematically correct. The use of this problem-solving strategy in the solution of the following problem is modeled in Examples A and B below. Example Problem A . Ima Hurryin is approaching a stoplight moving with a velocity of +30.0 m/s. The light turns yellow, and Ima applies the brakes and skids to a stop. If Ima's acceleration is -8.00 m/s2, then determine the displacement of the car during the skidding process. (Note that the direction of the velocity and the acceleration vectors are denoted by a + and a - sign.) The solution to this problem begins by the construction of an informative diagram of the physical situation. This is shown below. The second step involves the identification and listing of known information in variable form. Note that the vf value can be inferred to be 0 m/s since Ima's car comes to a stop. The initial velocity (vi) of the car is +30.0 m/s since this is the velocity at the beginning of the motion (the skidding motion). And the acceleration (a) of the car is given as - 8.00 m/s2. (Always pay careful attention to the + and - signs for the given quantities.) The next step of the strategy involves the listing of the unknown (or desired) information in variable form. In this case, the problem requests information about the displacement of the car. So d is the unknown quantity. The results of the first three steps are shown in the table below. Diagram: Given: Find: vi = +30.0 m/s vf = 0 m/s a = - 8.00 m/s2 d = ?? The next step of the strategy involves identifying a kinematic equation that would allow you to determine the unknown quantity. There are four kinematic equations to choose from. In general, you will always choose the equation that contains the three known and the one unknown variable. In this specific case, the three known variables and the one unknown variable are vf, vi, a, and d. Thus, you will look for an equation that has these four variables listed in it. An inspection of the four equations above reveals that the equation on the top right contains all four variables. vf2 = vi2 + 2 ⢠a ⢠d Once the equation is identified and written down, the next step of the strategy involves substituting known values into the equation and using proper algebraic steps to solve for the unknown information. This step is shown below. (0 m/s)2 = (30.0 m/s)2 + 2 ⢠(-8.00 m/s2) ⢠d 0 m2/s2 = 900 m2/s2 + (-16.0 m/s2) ⢠d (16.0 m/s2) ⢠d = 900 m2/s2 - 0 m2/s2 (16.0 m/s2)*d = 900 m2/s2 d = (900 m2/s2)/ (16.0 m/s2) d = (900 m2/s2)/ (16.0 m/s2) d = 56.3 m The solution above reveals that the car will skid a distance of 56.3 meters. (Note that this value is rounded to the third digit.) The last step of the problem-solving strategy involves checking the answer to assure that it is both reasonable and accurate. The value seems reasonable enough. It takes a car a considerable distance to skid from 30.0 m/s (approximately 65 mi/hr) to a stop. The calculated distance is approximately one-half a football field, making this a very reasonable skidding distance. Checking for accuracy involves substituting the calculated value back into the equation for displacement and insuring that the left side of the equation is equal to the right side of the equation. Indeed it is! Example Problem B Ben Rushin is waiting at a stoplight. When it finally turns green, Ben accelerated from rest at a rate of a 6.00 m/s2 for a time of 4.10 seconds. Determine the displacement of Ben's car during this time period. Once more, the solution to this problem begins by the construction of an informative diagram of the physical situation. This is shown below. The second step of the strategy involves the identification and listing of known information in variable form. Note that the vi value can be inferred to be 0 m/s since Ben's car is initially at rest. The acceleration (a) of the car is 6.00 m/s2. And the time (t) is given as 4.10 s. The next step of the strategy involves the listing of the unknown (or desired) information in variable form. In this case, the problem requests information about the displacement of the car. So d is the unknown information. The results of the first three steps are shown in the table below. Diagram: Given: Find: vi = 0 m/s t = 4.10 s a = 6.00 m/s2 d = ?? The next step of the strategy involves identifying a kinematic equation that would allow you to determine the unknown quantity. There are four kinematic equations to choose from. Again, you will always search for an equation that contains the three known variables and the one unknown variable. In this specific case, the three known variables and the one unknown variable are t, vi, a, and d. An inspection of the four equations above reveals that the equation on the top left contains all four variables. d = vi ⢠t + ½ ⢠a ⢠t2 Once the equation is identified and written down, the next step of the strategy involves substituting known values into the equation and using proper algebraic steps to solve for the unknown information. This step is shown below. d = (0 m/s) ⢠(4.1 s) + ½ ⢠(6.00 m/s2) ⢠(4.10 s)2 d = (0 m) + ½ ⢠(6.00 m/s2) ⢠(16.81 s2) d = 0 m + 50.43 m d = 50.4 m The solution above reveals that the car will travel a distance of 50.4 meters. (Note that this value is rounded to the third digit.) The last step of the problem-solving strategy involves checking the answer to assure that it is both reasonable and accurate. The value seems reasonable enough. A car with an acceleration of 6.00 m/s/s will reach a speed of approximately 24 m/s (approximately 50 mi/hr) in 4.10 s. The distance over which such a car would be displaced during this time period would be approximately one-half a football field, making this a very reasonable distance. Checking for accuracy involves substituting the calculated value back into the equation for displacement and insuring that the left side of the equation is equal to the right side of the equation. Indeed, it is! The two example problems above illustrate how the kinematic equations can be combined with a simple problem-solving strategy to predict unknown motion parameters for a moving object. Provided that three motion parameters are known, any of the remaining values can be determined. In the next part of Lesson 6, we will see how this strategy can be applied to free fall situations. Or if interested, you can try some practice problems and check your answer against the given solutions. Kinematic Equations and Free Fall As mentioned in Lesson 5, a free-falling object is an object that is falling under the sole influence of gravity. That is to say that any object that is moving and being acted upon only be the force of gravity is said to be "in a state of free fall." Such an object will experience a downward acceleration of 9.8 m/s/s. Whether the object is falling downward or rising upward towards its peak, if it is under the sole influence of gravity, then its acceleration value is 9.8 m/s/s. Like any moving object, the motion of an object in free fall can be described by four kinematic equations. The kinematic equations that describe any object's motion are: The symbols in the above equation have a specific meaning: the symbol d stands for the displacement; the symbol t stands for the time; the symbol a stands for the acceleration of the object; the symbol vi stands for the initial velocity value; and the symbol vf stands for the final velocity. Applying Free Fall Concepts to Problem-Solving There are a few conceptual characteristics of free fall motion that will be of value when using the equations to analyze free fall motion. These concepts are described as follows: ⢠An object in free fall experiences an acceleration of -9.8 m/s/s. (The - sign indicates a downward acceleration.) Whether explicitly stated or not, the value of the acceleration in the kinematic equations is -9.8 m/s/s for any freely falling object. ⢠If an object is merely dropped (as opposed to being thrown) from an elevated height, then the initial velocity of the object is 0 m/s. ⢠If an object is projected upwards in a perfectly vertical direction, then it will slow down as it rises upward. The instant at which it reaches the peak of its trajectory, its velocity is 0 m/s. This value can be used as one of the motion parameters in the kinematic equations; for example, the final velocity (vf) after traveling to the peak would be assigned a value of 0 m/s. ⢠If an object is projected upwards in a perfectly vertical direction, then the velocity at which it is projected is equal in magnitude and opposite in sign to the velocity that it has when it returns to the same height. That is, a ball projected vertically with an upward velocity of +30 m/s will have a downward velocity of -30 m/s when it returns to the same height. These four principles and the four kinematic equations can be combined to solve problems involving the motion of free-falling objects. The two examples below illustrate application of free fall principles to kinematic problem-solving. In each example, the problem solving strategy that was introduced earlier in this lesson will be utilized. Example Problem A Luke Autbeloe drops a pile of roof shingles from the top of a roof located 8.52 meters above the ground. Determine the time required for the shingles to reach the ground. The solution to this problem begins by the construction of an informative diagram of the physical situation. This is shown below. The second step involves the identification and listing of known information in variable form. You might note that in the statement of the problem, there is only one piece of numerical information explicitly stated: 8.52 meters. The displacement (d) of the shingles is -8.52 m. (The - sign indicates that the displacement is downward). The remaining information must be extracted from the problem statement based upon your understanding of the above principles. For example, the vi value can be inferred to be 0 m/s since the shingles are dropped (released from rest; see note above). And the acceleration (a) of the shingles can be inferred to be -9.8 m/s2 since the shingles are free-falling (see note above). (Always pay careful attention to the + and - signs for the given quantities.) The next step of the solution involves the listing of the unknown (or desired) information in variable form. In this case, the problem requests information about the time of fall. So t is the unknown quantity. The results of the first three steps are shown in the table below. Diagram: Given: Find: vi = 0.0 m/s d = -8.52 m a = - 9.8 m/s2 t = ?? The next step involves identifying a kinematic equation that allows you to determine the unknown quantity. There are four kinematic equations to choose from. In general, you will always choose the equation that contains the three known and the one unknown variable. In this specific case, the three known variables and the one unknown variable are d, vi, a, and t. Thus, you will look for an equation that has these four variables listed in it. An inspection of the four equations above reveals that the equation on the top left contains all four variables. d = vi ⢠t + ½ ⢠a ⢠t2 Once the equation is identified and written down, the next step involves substituting known values into the equation and using proper algebraic steps to solve for the unknown information. This step is shown below. -8.52 m = (0 m/s) ⢠(t) + ½ ⢠(-9.8 m/s2) ⢠(t)2 -8.52 m = (0 m) *(t) + (-4.9 m/s2) ⢠(t)2 -8.52 m = (-4.9 m/s2) ⢠(t)2 (-8.52 m)/(-4.9 m/s2) = t2 1.739 s2 = t2 t = 1.32 s The solution above reveals that the shingles will fall for a time of 1.32 seconds before hitting the ground. (Note that this value is rounded to the third digit.) The last step of the problem-solving strategy involves checking the answer to assure that it is both reasonable and accurate. The value seems reasonable enough. The shingles are falling a distance of approximately 10 yards (1 meter is pretty close to 1 yard); it seems that an answer between 1 and 2 seconds would be highly reasonable. The calculated time easily falls within this range of reasonability. Checking for accuracy involves substituting the calculated value back into the equation for time and insuring that the left side of the equation is equal to the right side of the equation. Indeed it is! Example Problem B Rex Things throws his mother's crystal vase vertically upwards with an initial velocity of 26.2 m/s. Determine the height to which the vase will rise above its initial height. Once more, the solution to this problem begins by the construction of an informative diagram of the physical situation. This is shown below. The second step involves the identification and listing of known information in variable form. You might note that in the statement of the problem, there is only one piece of numerical information explicitly stated: 26.2 m/s. The initial velocity (vi) of the vase is +26.2 m/s. (The + sign indicates that the initial velocity is an upwards velocity). The remaining information must be extracted from the problem statement based upon your understanding of the above principles. Note that the vf value can be inferred to be 0 m/s since the final state of the vase is the peak of its trajectory (see note above). The acceleration (a) of the vase is -9.8 m/s2 (see note above). The next step involves the listing of the unknown (or desired) information in variable form. In this case, the problem requests information about the displacement of the vase (the height to which it rises above its starting height). So d is the unknown information. The results of the first three steps are shown in the table below. Diagram: Given: Find: vi = 26.2 m/s vf = 0 m/s a = -9.8 m/s2 d = ?? The next step involves identifying a kinematic equation that would allow you to determine the unknown quantity. There are four kinematic equations to choose from. Again, you will always search for an equation that contains the three known variables and the one unknown variable. In this specific case, the three known variables and the one unknown variable are vi, vf, a, and d. An inspection of the four equations above reveals that the equation on the top right contains all four variables. vf2 = vi2 + 2 ⢠a ⢠d Once the equation is identified and written down, the next step involves substituting known values into the equation and using proper algebraic steps to solve for the unknown information. This step is shown below. (0 m/s)2 = (26.2 m/s)2 + 2 â˘(-9.8m/s2) â˘d 0 m2/s2 = 686.44 m2/s2 + (-19.6 m/s2) â˘d (-19.6 m/s2) ⢠d = 0 m2/s2 -686.44 m2/s2 (-19.6 m/s2) ⢠d = -686.44 m2/s2 d = (-686.44 m2/s2)/ (-19.6 m/s2) d = 35.0 m The solution above reveals that the vase will travel upwards for a displacement of 35.0 meters before reaching its peak. (Note that this value is rounded to the third digit.) The last step of the problem-solving strategy involves checking the answer to assure that it is both reasonable and accurate. The value seems reasonable enough. The vase is thrown with a speed of approximately 50 mi/hr (merely approximate 1 m/s to be equivalent to 2 mi/hr). Such a throw will never make it further than one football field in height (approximately 100 m), yet will surely make it past the 10-yard line (approximately 10 meters). The calculated answer certainly falls within this range of reasonability. Checking for accuracy involves substituting the calculated value back into the equation for displacement and insuring that the left side of the equation is equal to the right side of the equation. Indeed, it is! Kinematic equations provide a useful means of determining the value of an unknown motion parameter if three motion parameters are known. In the case of a free-fall motion, the acceleration is often known. And in many cases, another motion parameter can be inferred through a solid knowledge of some basic kinematic principles.
To understand melody in music, think about some music youâre familiar with. If you were asked to hum it, what would that sound like? The part of the music that youâd hum is the melody. Itâs the main thread of sound that your brain tracks and holds onto when youâre listening to music. In vocal music, the melody is sung by the lead singer. Other vocalists can provide harmony and instruments can add accompaniment, but the melody is the star of the show.What are the characteristics of melody in music? How do you describe a melody in music? A melody needs to have two things. The first is a sequence of notes, or pitches, which range from high to low. The second is rhythm, which is the timing and duration of each note. These two simple elements can create an incredible variety of combinations. Even though a melody only consists of one note at a time, it can convey so much energy and emotion. Melodies can be fast and sparkly, like âThe Flight of the Bumblebee.â They can be slow and majestic, like âFinlandia.â They might be sweeping and graceful, like a Strauss waltz. Or they can be fun and exciting, like your favorite pop tunes that you love to sing along with. Melodies often tell you a lot about where a piece of music comes from. Itâs easy to recognize and identify melodies from different folk traditions such as the Japanese folk song âSakuraâ or the Irish tune âStar of the County Down.â Learn how to play your favorite melodies on piano, and more! Sign up now. What is melody in music? Here are some examples. Here is the famous melody for the song âLean on Meâ written out on a staff. Notice the way that the notes move up, down, and then repeat. What is melody in music? Example of Lean On Me notes on treble staff. A melody all by itself is great, but music can be even more fun when thereâs an accompaniment. Here are a few bars of âLean on Meâ with the accompaniment written out. As you listen to this song, notice how the accompaniment has a very similar rhythm and movement to the melody. Then thereâs that one note in the bass line that comes along every measure with its own rhythm, which adds some extra energy and movement to the song. What makes a good melody? When you create a melody, there are four types of movement you can use: Repeat (same note) Step (up or down) Skip (up or down) Leap (up or down) Stepping and repeating are the most common types of melodic motion, and this makes a melody easier to sing. Most âhummableâ tunes use steps and repeats almost exclusively. This kind of melody is called conjunct. Beethovenâs âOde to Joy,â one of the most famous melodies of all time.Skips and leaps are generally more sparing in melodies, but when thoughtfully placed they can have a powerful emotional impact. Tunes with a lot of leaps are called disjunct. Listen to Sarah Brightman sing All I Ask of You from The Phantom of the Opera starting at 0:39. This is a very disjunct melody, and challenging to sing. Great melodies also incorporate patterns that blend unity, repetition, and contrast. Our ears love patterns, but they also love novelty and growth. A good melody incorporates all of these elements. For example, listen to John Williamâs âPrincess Leia Theme.â Can you hear the repeated pattern in the melody that gradually moves higher as the theme progresses? Now listen to the way it changes and develops into something that fits with what came before but sounds new at the same time. This is some great melodic writing! Can melody exist without rhythm? There is no way for a melody to exist without rhythm. Even if your melody only has one note, that note has a duration, and thatâs the rhythm. If your melody has two notes, how long those notes last and how much time passes between hearing them is also a rhythm. A melody in music can often be recognized even when itâs performed with different rhythms. This frequently happens in live performances of pop, rock, and jazz, in which singers typically improvise slight rhythmic differences with each performance. No two renditions are exactly the same, and this constant reinterpretation keeps the music fresh. How to make a melody for a song on piano Creating your own melodies on the piano is easy and fun! There are so many ways you can discover a melody all your own. Here are a few ideas. Get some inspiration from the world around you. What can you hear right now? A clock ticking? A bird song? A car passing by your house? See if you can find some notes on the piano that imitate the sounds you hear. Think of a feeling youâd like to put into a melody. What are some ways you could make a string of notes sound happy, sad, angry, or maybe just thoughtful. Choose a line from a poem you like, or write your own. Read it out loud and put some feeling into it. Did your voice rise and fall in pitch as you were reading? Now go to the piano, start on any note you like, and try to imitate what happened when you read. Go up when your voice naturally went up, go down when your voice naturally went down. How did that sound? Now you have the perfect melody to go with those words. Too many keys on the piano? The truth is, most melodies use only a limited number of different notes. Try creating a melody using only the black keys. These form whatâs called a pentatonic scale. Itâs used in a lot of folk music traditions around the world and can be a great place to start if you want to create your own melodies. Remember, when you create your melody, keep it simple. Use repeated notes and steps, but add a few skips to keep things interesting. One tip about leaps: when you do put in a big leap, try doubling back and filling in the empty space you leaped over. This keeps the melody self-contained and easier to sing. Also, see if you can use the same patterns of notes and rhythms to give the melody unity, but also change those patterns to give it variety. There is no right or wrong way to create your own music. Keep trying combinations of notes and rhythms until you find something that you like. How many bars and notes are in a melody? Many types of music tend to have a prescribed number of bars, or measures. This will vary widely between different genres, and creates an overall sense of musical structure. If youâre writing a pop song, a verse will usually have between eight and sixteen bars. The prechorus that follows often has just four bars, and this âforeshorteningâ creates a sense of acceleration, driving the listener toward the chorus. The number of notes can also vary widely. A melody in music needs at least two notes, and a long and complex one can have hundreds or even thousands of notes. What is a countermelody in music? How many melodies should a song have? A counter melody is a melodic line that interacts with the primary melody as an independent but supportive voice. A great example of this is the song âWe Donât Talk about Bruno.â Each character sings their own melody during the piece, but these melodies all combine at the end as countermelodies. This produces a musical texture known as counterpoint. The same thing happens in âOne Day Moreâ from Les Miserables. The different melodies are first sung separately, but end up being combined in a splendid, complex texture that leads the music to its thrilling conclusion. The difference between a countermelody and regular harmony is that harmony usually supports the rhythms of the melody. A countermelody will move more independently, with different rhythms from those of the melody, and will often sound âmelodicâ when sung or played all by itself. A melodic song should have one main melody. This is the part that the lead voice sings. Itâs usually in the spotlight, and will be the most memorable part of the music. Anything else is either harmony, countermelody, or accompaniment. Does all music have to have a melody? A piece of music doesnât have to have a melody. There are many different kinds of music without melody. For example, a lot of music played on percussion instruments wonât have a melody. Listen to this example of Tahitian drumming. This is some great music, exciting and fun to listen to, but youâd have a hard time humming it. Itâs music, but it doesnât have a melody. Rap music is another style of music where there doesnât have to be a melody. In rap, words are chanted rather than sung. The performer will raise and lower the pitch of their voice for emphasis, but itâs the rhythm of the words that creates most of the music. Music can even lack any melody, at least in some sections. Listen to the opening chords of âDuel of the Fates.â This choral passage is all about harmony, with little rhythmic variance or sense of melody. But it makes an effective contrast with the next section, which is bustling with rapid instrumental melodies. In some pieces, there are multiple melodic lines but there is no one main melody. When music is made up of equally important countermelodies, it creates a contrapuntal texture. Baroque composer J.S. Bach was one of the greatest masters of this style, such as in his Little Fugue in G minor. It starts with a single melodic line, the subject, but then a countermelody is added, and then more and more until several melodic lines are playing together. Itâs fun to listen to, but once all the countermelodies are playing together it becomes hard to decide which part to hum along with! Youâll also hear a lot of counterpoint in jazz music, in which the different instruments are all playing together and improvising their own melodies that combine to create a rich, thick musical texture. Experience the wonder of melody in music! Whether youâre humming your favorite tune, or creating a new song all your own, melody is a memorable, shareable part of music. Enrich your music experience by being aware of, listening for, and enjoying the melodies all around you.
She went by the name of Belisa Crepusculario, not because she had been baptized with that name or given it by her mother, but because she herself had searched until she found the poetry of "beauty" and "twilight" and cloaked herself in it. She made her living selling words. She journeyed through the country from the high cold mountains to the burning coasts, stopping at fairs and in markets where she set up four poles covered by a canvas awning under which she took refuge from the sun and rain to minister to her customers. She did not have to peddle her merchandise because from having wandered far and near, everyone knew who she was. Some people waited for her from one year to the next, and when she appeared in the village with her bundle beneath her arm, they would form a line in front of her stall. Her prices were fair. For five centavos she delivered verses from memory, for seven she improved the quality of dreams, for nine she wrote love letters, for twelve she invented insults for irreconcilable enemies. She also sold stories, not fantasies but long, true stories she recited at one telling, never skipping a word. This is how she carried news from one town to another. People paid her to add a line or two: our son was born, so-and-so died, our children got married, the crops burned in the field. Wherever she went a small crowd gathered around to listen as she began to speak, and that was how they learned about each others' doings, about distant relatives, about what was going on in the civil war. To anyone who paid her fifty centavos in trade, she gave the gift of a secret word to drive away melancholy. It was not the same word for everyone, naturally, because that would have been collective dece it. Each person received his or her own word, with the assurance that no one else would use it that way in this universe or the Beyond. Belisa Crepusculario had been born into a family so poor they did not even have names to give their children. She came into the world and grew up in an inhospitable land where some years the rains became avalanches of water that bore everything away before them and others when not a drop fell from the sky and the sun swelled to fill the horizon and the world became a desert. Until she was twelve, Belisa had no occupation or virtue other than having withstood hunger and the exhaustion of centuries. During one interminable drought, it fell to her to bury four younger brothers and sisters, when she realized that her turn was next, she decided to set out across the 2 plains in the direction of the sea, in hopes that she might trick death along the way. The land was eroded, split with deep cracks, strewn with rocks, fossils of trees and thorny bushes, and skeletons of animals bleached by the sun. From time to time she ran into families who, like her, were heading south, following the mirage of water. Some had begun the march carrying their belongings on their back or in small carts, but they could barely move their own bones, and after a while they had to abandon their possessions. They dragged themselves along painfully, their skin turned to lizard hide and their eyes burned by the reverberating glare. Belisa greeted them with a wave as she passed, but she did not stop, because she had no strength to waste in acts of compassion. Many people fell by the wayside, but she was so stubborn that she survived to cross through that hell and at long last reach the first trickles of water, fine, almost invisible threads that fed spindly vegetation and farther down widened into small streams and marshes. Belisa Crepusculario saved her life and in the process accidentally discovered writing. In a village near the coast, the wind blew a page of newspaper at her feet. She picked up the brittle yellow paper and stood a long while looking at it, unable to determine its purpose, until curiosity overcame her shyness. She walked over to a man who was washing his horse in the muddy pool where she had quenched her thirst. "What is this?" she asked. "The sports page of the newspaper," the man replied, concealing his surprise at her ignorance. The answer astounded the girl, but she did not want to seem rude, so she merely inquired about the significance of the fly tracks scattered across the page. "Those are words, child. Here it says that Fulgencio Barba knocked out El Negro Tiznao in the third round." That was the day Belisa Crepusculario found out that words make their way in the world without a master, and that anyone with a little cleverness can appropriate them and do business with them. She made a quick assessment of her situation and concluded that aside from becoming a prostitute or working as a servant in the kitchens of the rich there were few occupations she was qualified for. It seemed to her that selling words would be an honorable alternative. From that moment on, she worked at that profession, and was never tempted by any other. At the beginning, she offered her merchandise unaware that words could be written outside of newspapers. When she learned otherwise, she calculated the infinite possibilities of her trade and with her savings paid a priest twenty pesos to teach her to read and write, with her three 3 remaining coins she bought a dictionary. She poured over it from A to Z and then threw it into the sea, because it was not her intention to defraud her customers with packaged words. One August morning several years later, Belisa Crepusculario was sitting in her tent in the middle of a plaza, surrounded by the uproar of market day, selling legal arguments to an old man who had been trying for sixteen years to get his pension. Suddenly she heard yelling and thudding hoofbeats. She looked up from her writing and saw, first, a cloud of dust, and then a band of horsemen come galloping into the plaza. They were the Colonel's men, sent under orders of El Mulato, a giant known throughout the land for the speed of his knife and his loyalty to his chief. Both the Colonel and El Mulato had spent their lives fighting in the civil war, and their names were ineradicably linked to devastation and calamity. The rebels swept into town like a stampeding herd, wrapped in noise, bathed in sweat, and leaving a hurricane of fear in their trail. Chickens took wing, dogs ran for their lives, women and children scurried out of sight, until the only living soul left in the market was Belisa Crepusculario. She had never seen El Mulato and was surprised to see him walking toward her. "I'm looking for you," he shouted, pointing his coiled whip at her, even before the words were out, two men rushed her -- knocking over her canopy and shattering her inkwell -- bound her hand and foot, and threw her like a sea bag across the rump of El Mulato's mount. Then they thundered off toward the hills. Hours later, just as Belisa Crepusculario was near death, her heart ground to sand by the pounding of the horse, they stopped, and four strong hands set her down. She tried to stand on her feet and hold her head high, but her strength failed her and she slumped to the ground, sinking into a confused dream. She awakened several hours later to the murmur of night in the camp, but before she had time to sort out the sounds, she opened her eyes and found herself staring into the impatient glare of El Mulato, kneeling beside her. "Well, woman, at last you've come to," he said. To speed her to her senses, he tipped his canteen and offered her a sip of liquor laced with gunpowder. She demanded to know the reason for such rough treatment, and El Mulato explained that the Colonel needed her services. He allowed her to splash water on her face, and then led her to the far end of the camp where the most feared man in all the land was lazing in a hammock strung between two trees. She could not see his face, because he lay in the deceptive shadow of the leaves and the indelible shadow of all his years as a bandit, but she imagined from the way his 4 gigantic aide addressed him with such humility that he must have a very menacing expression. She was surprised by the Colonel's voice, as soft and well-modulated as a professor's. "Are you the woman who sells words?" he asked. "At your service," she stammered, peering into the dark and trying to see him better. The Colonel stood up, and turned straight toward her. She saw dark skin and the eyes of a ferocious puma, and she knew immediately that she was standing before the loneliest man in the world. "I want to be President," he announced. The Colonel was weary of riding across that godforsaken land, waging useless wars and suffering defeats that no subterfuge could transform into victories. For years he had been sleeping in the open air, bitten by mosquitoes, eating iguanas and snake soup, but those minor inconveniences were not why he wanted to change his destiny. What truly troubled him was the terror he saw in people's eyes. He longed to ride into a town beneath a triumphal arch with bright flags and flowers everywhere, he wanted to be cheered, and be given newly laid eggs and freshly baked bread. Men fled at the sight of him, children trembled, and women miscarried from fright, he had had enough, and so he had decided to become President. El Mulato had suggested that they ride to the capital, gallop up to the Palace, and take over the government, the way they had taken so many other things without anyone's permission. The Colonel, however, did not want to be just another tyrant, there had been enough of those before him and, besides, if he did that, he would never win people's hearts. It was his aspiration to win the popular vote in the December elections. "To do that, I have to talk like a candidate. Can you sell me the words for a speech?" the Colonel asked Belisa Crepusculario. She had accepted many assignments, but none like this. She did not dare refuse, fearing that El Mulato would shoot her between the eyes, or worse still, that the Colonel would burst into tears. There was more to it than that, however, she felt the urge to help him because she felt a throbbing warmth beneath her skin, a powerful desire to touch that man, to fondle him, to clasp him in her arms. All night and a good part of the following day, Belisa Crepusculario searched her repertory for words adequate for a presidential speech, closely watched by El Mulato, who could not take his eyes from her firm wanderer's legs and virginal breasts. She discarded harsh, cold words, words 5 that were too flowery, words worn from abuse, words that offered improbable promises, untruthful and confusing words, until all she had left were words sure to touch the minds of men and women's intuition. Calling upon the knowledge she had purchased from the priest for twenty pesos, she wrote the speech on a sheet of paper and then signaled El Mulato to untie the rope that bound her ankles to a tree. He led her once more to the Colonel, and again she felt the throbbing anxiety that had seized her when she first saw him. She handed him the paper and waited while he looked at it, holding it gingerly between thumbs and fingertips. "What the shit does this say," he asked finally. "Don't you know how to read?" "War's what I know," he replied. She read the speech aloud. She read it three times, so her client could engrave it on his memory. When she finished, she saw the emotion in the faces of the soldiers who had gathered round to listen, and saw that the Colonel's eyes glittered with enthusiasm, convinced that with those words the presidential chair would be his. "If after they've heard it three times, the boys are still standing there with their mouths hanging open, it must mean the thing's damn good, Colonel" was El Mulato's approval. "All right, woman. How much do I owe you?" the leader asked. "One peso, Colonel." "That's not much," he said, opening the pouch he wore at his belt, heavy with proceeds from the last foray. "The peso entitles you to a bonus. I'm going to give you two secret words," said Belisa Crepusculario. "What for?" She explained that for every fifty centavos a client paid, she gave him the gift of a word for his exclusive use. The Colonel shrugged. He had no interest at all in her offer, but he did not want to be impolite to someone who had served him so well. She walked slowly to the leather stool where he was sitting, and bent down to give him her gift. The man smelled the scent of a mountain cat issuing from the woman, a fiery heat radiating from her hips, he heard the terrible whisper of her hair, and a breath of sweetmint murmured into his ear the two secret words that were his alone. "They are yours, Colonel," she said as she stepped back. "You may use them as much as you 6 please." El Mulato accompanied Belisa to the roadside, his eyes as entreating as a stray dog's, but when he reached out to touch her, he was stopped by an avalanche of words he had never heard before; believing them to be an irrevocable curse, the flame of his desire was extinguished. During the months of September, October, and November the Colonel delivered his speech so many times that had it not been crafted from glowing and durable words it would have turned to ash as he spoke. He travelled up and down and across the country, riding into cities with a triumphal air, stopping in even the most forgotten villages where only the dump heap betrayed a human presence, to convince his fellow citizens to vote for him. While he spoke from a platform erected in the middle of the plaza, El Mulato and his men handed out sweets and painted his name on all the walls in gold frost. No one paid the least attention to those advertising ploys; they were dazzled by the clarity of the Colonel's proposals and the poetic lucidity of his arguments, infected by his powerful wish to right the wrongs of history, happy for the first time in their lives. When the Candidate had finished his speech, his soldiers would fire their pistols into the air and set off firecrackers, and when finally they rode off, they left behind a wake of hope that lingered for days on the air, like the splendid memory of a comet's tail. Soon the Colonel was the favorite. No one had ever witnessed such a phenomenon: a man who surfaced from the civil war, covered with scars and speaking like a professor, a man whose fame spread to every corner of the land and captured the nation's heart. The press focused their attention on him. Newspapermen came from far away to interview him and repeat his phrases, and the number of his followers and enemies continued to grow. "We're doing great, Colonel," said El Mulato, after twelve successful weeks of campaigning. But the Candidate did not hear. He was repeating his secret words, as he did more and more obsessively. He said them when he was mellow with nostalgia; he murmured them in his sleep; he carried them with him on horseback; he thought them before delivering his famous speech; and he caught himself savoring them in his leisure time. And every time he thought of those two words, he thought of Belisa Crepusculario, and his senses were inflamed with the memory of her feral scent, her fiery heat, the whisper of her hair, and her sweetmint breath in his ear, until he began to go around like a sleepwalker, and his men realized that he might die before he ever sat in the presidential chair. "What's got hold of you, Colonel," El Mulato asked so often that finally one day his chief broke 7 down and told him the source of his befuddlement: those two words that were buried like two daggers in his gut. "Tell me what they are and maybe they'll lose their magic," his faithful aide suggested. "I can't tell them, they're for me alone," the Colonel replied. Saddened by watching his chief decline like a man with a death sentence on his head, El Mulato slung his rifle over his shoulder and set out to find Belisa Crepusculario. He followed her trail through all that vast country, until he found her in a village in the far south, sitting under her tent reciting her rosary of news. He planted himself, spraddle-legged, before her, weapon in hand. "You! You're coming with me," he ordered. She had been waiting. She picked up her inkwell, folded the canvas of her small stall, arranged her shawl around her shoulders, and without a word took her place behind El Mulato's saddle. They did not exchange so much as a word in all the trip; El Mulato's desire for her had turned into rage, and only his fear of her tongue prevented his cutting her to shreds with his whip. Nor was he inclined to tell her that the Colonel was in a fog, and that a spell whispered into his ear had done what years of battle had not been able to do. Three days later they arrived at the encampment, and immediately, in view of all the troops, El Mulato led his prisoner before the Candidate. "I brought this witch here so you can give her back her words, Colonel," El Mulato said, pointing the barrel of his rifle at the woman's head. "And then she can give you back your manhood." The Colonel and Belisa Crepusculario stared at each other, measuring one another from a distance. The men knew then that their leader would never undo the witchcraft of those accursed words, because the whole world could see the voracious-puma eyes soften as the woman walked to him and took his hand in hers. Copyright Š 1989 by Isabel Allende From The Stories of Eva Luna, Translated by Margaret Sayers Peden
Writing Numbers in Word Form - Starter Quiz