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Writing and solving linear equations from real world situations
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Writing and Solving 1 and 2 Step Equations2.3.5 Writing and Solving Inequalities Solve real-world and mathematical problems by writing and solving equations of the form: • 𝑥 + 𝑝 = 𝑞 in which p, q and x are all nonnegative rational numbers; and, • 𝑝 ∙ 𝑥 = 𝑞 for cases in which p, q and x are all nonnegative rational numbersModeling, Writing and Solving Equations and Inequalities with Variables on Both Sides ILLINOIS PROFESSIONAL TEACHING STANDARDS (2013) Standard 1 - Teaching Diverse Students – The competent teacher understands the diverse characteristics and abilities of each student and how individuals develop and learn within the context of their social, economic, cultural, linguistic, and academic experiences. The teacher uses these experiences to create instructional opportunities that maximize student learning. Knowledge Indicators – The competent teacher: 1A) understands the spectrum of student diversity (e.g., race and ethnicity, socioeconomic status, special education, gifted, English language learners (ELL), sexual orientation, gender, gender identity) and the assets that each student brings to learning across the curriculum; 1B) understands how each student constructs knowledge, acquires skills, and develops effective and efficient critical thinking and problem-solving capabilities; 1C) understands how teaching and student learning are influenced by development (physical, social and emotional, cognitive, linguistic), past experiences, talents, prior knowledge, economic circumstances and diversity within the community; 1D) understands the impact of cognitive, emotional, physical, and sensory disabilities on learning and communication pursuant to the Individuals with Disabilities Education Improvement Act (also referred to as “IDEA”) (20 USC 1400 et seq.), its implementing regulations (34 CFR 300; 2006), Article 14 of the School Code [105 ILCS 5/Art.14] and 23 Ill. Adm. Code 226 (Special Education); 1E) understands the impact of linguistic and cultural diversity on learning and communication; 1F) understands his or her personal perspectives and biases and their effects on one’s teaching; and 1G) understands how to identify individual needs and how to locate and access technology, services, and resources to address those needs. Performance Indicators – The competent teacher: 1H) analyzes and uses student information to design instruction that meets the diverse needs of students and leads to ongoing growth and achievement; 1I) stimulates prior knowledge and links new ideas to already familiar ideas and experiences; 1J) differentiates strategies, materials, pace, levels of complexity, and language to introduce concepts and principles so that they are meaningful to students at varying levels of development and to students with diverse learning needs; 1K) facilitates a learning community in which individual differences are respected; and 1L) uses information about students’ individual experiences, families, cultures, and communities to create meaningful learning opportunities and enrich instruction for all students. Standard 2 - Content Area and Pedagogical Knowledge – The competent teacher has in-depth understanding of content area knowledge that includes central concepts, methods of inquiry, structures of the disciplines, and content area literacy. The teacher creates meaningful learning experiences for each student based upon interactions among content area and pedagogical knowledge, and evidence-based practice. Knowledge Indicators – The competent teacher: 2A) understands theories and philosophies of learning and human development as they relate to the range of students in the classroom; 2B) understands major concepts, assumptions, debates, and principles; processes of inquiry; and theories that are central to the disciplines; 2C) understands the cognitive processes associated with various kinds of learning (e.g., critical and creative thinking, problem-structuring and problem-solving, invention, memorization, and recall) 2 and ensures attention to these learning processes so that students can master content standards; 2D) understands the relationship of knowledge within the disciplines to other content areas and to life applications; 2E) understands how diverse student characteristics and abilities affect processes of inquiry and influence patterns of learning; 2F) knows how to access the tools and knowledge related to latest findings (e.g., research, practice, methodologies) and technologies in the disciplines; 2G) understands the theory behind and the process for providing support to promote learning when concepts and skills are first being introduced; and 2H) understands the relationship among language acquisition (first and second), literacy development, and acquisition of academic content and skills. Performance Indicators – The competent teacher: 2I) evaluates teaching resources and materials for appropriateness as related to curricular content and each student’s needs; 2J) uses differing viewpoints, theories, and methods of inquiry in teaching subject matter concepts; 2K) engages students in the processes of critical thinking and inquiry and addresses standards of evidence of the disciplines; 2L) demonstrates fluency in technology systems, uses technology to support instruction and enhance student learning, and designs learning experiences to develop student skills in the application of technology appropriate to the disciplines; 2M) uses a variety of explanations and multiple representations of concepts that capture key ideas to help each student develop conceptual understanding and address common misunderstandings; 2N) facilitates learning experiences that make connections to other content areas and to life experiences; 2O) designs learning experiences and utilizes assistive technology and digital tools to provide access to general curricular content to individuals with disabilities; 2P) adjusts practice to meet the needs of each student in the content areas; and 2Q) applies and adapts an array of content area literacy strategies to make all subject matter accessible to each student. Standard 3 - Planning for Differentiated Instruction – The competent teacher plans and designs instruction based on content area knowledge, diverse student characteristics, student performance data, curriculum goals, and the community context. The teacher plans for ongoing student growth and achievement. Knowledge Indicators – The competent teacher: 3A) understands the Illinois Learning Standards (23 Ill. Adm. Code 1.Appendix D), curriculum development process, content, learning theory, assessment, and student development and knows how to incorporate this knowledge in planning differentiated instruction; 3B) understands how to develop short- and long-range plans, including transition plans, consistent with curriculum goals, student diversity, and learning theory; 3C) understands cultural, linguistic, cognitive, physical, and social and emotional differences, and considers the needs of each student when planning instruction; 3D) understands when and how to adjust plans based on outcome data, as well as student needs, goals, and responses; 3E) understands the appropriate role of technology, including assistive technology, to address student needs, as well as how to incorporate contemporary tools and resources to maximize student learning; 3 3F) understands how to co-plan with other classroom teachers, parents or guardians, paraprofessionals, school specialists, and community representatives to design learning experiences; and 3G) understands how research and data guide instructional planning, delivery, and adaptation. Performance Indicators – The competent teacher: 3H) establishes high expectations for each student’s learning and behavior; 3I) creates short-term and long-term plans to achieve the expectations for student learning; 3J) uses data to plan for differentiated instruction to allow for variations in individual learning needs; 3K) incorporates experiences into instructional practices that relate to a student’s current life experiences and to future life experiences; 3L) creates approaches to learning that are interdisciplinary and that integrate multiple content areas; 3M) develops plans based on student responses and provides for different pathways based on student needs; 3N) accesses and uses a wide range of information and instructional technologies to enhance a student’s ongoing growth and achievement; 3O) when planning instruction, addresses goals and objectives contained in plans developed under Section 504 of the Rehabilitation Act of 1973 (29 USC 794), individualized education programs (IEP) (see 23 Ill. Adm. Code 226 (Special Education)) or individual family service plans (IFSP) (see 23 Ill. Adm. Code 226 and 34 CFR 300.24; 2006); 3P) works with others to adapt and modify instruction to meet individual student needs; and 3Q) develops or selects relevant instructional content, materials, resources, and strategies (e.g., project-based learning) for differentiating instruction. Standard 4 - Learning Environment – The competent teacher structures a safe and healthy learning environment that facilitates cultural and linguistic responsiveness, emotional well-being, self-efficacy, positive social interaction, mutual respect, active engagement, academic risk-taking, self-motivation, and personal goal-setting. Knowledge Indicators – The competent teacher: 4A) understands principles of and strategies for effective classroom and behavior management; 4B) understands how individuals influence groups and how groups function in society; 4C) understands how to help students work cooperatively and productively in groups; 4D) understands factors (e.g., self-efficacy, positive social interaction) that influence motivation and engagement; 4E) knows how to assess the instructional environment to determine how best to meet a student’s individual needs; 4F) understands laws, rules, and ethical considerations regarding behavior intervention planning and behavior management (e.g., bullying, crisis intervention, physical restraint); 4G) knows strategies to implement behavior management and behavior intervention planning to ensure a safe and productive learning environment; and 4H) understands the use of student data (formative and summative) to design and implement behavior management strategies. Performance Indicators – The competent teacher: 4I) creates a safe and healthy environment that maximizes student learning; 4J) creates clear expectations and procedures for communication and behavior and a physical setting conducive to achieving classroom goals; 4K) uses strategies to create a smoothly functioning learning community in which students assume responsibility for themselves and one another, participate in decision-making, work collaboratively and independently, use appropriate technology, and engage in purposeful learning activities; 4 4L) analyzes the classroom environment and makes decisions to enhance cultural and linguistic responsiveness, mutual respect, positive social relationships, student motivation, and classroom engagement; 4M) organizes, allocates, and manages time, materials, technology, and physical space to provide active and equitable engagement of students in productive learning activities; 4N) engages students in and monitors individual and group-learning activities that help them develop the motivation to learn; 4O) uses a variety of effective behavioral management techniques appropriate to the needs of all students that include positive behavior interventions and supports; 4P) modifies the learning environment (including the schedule and physical arrangement) to facilitate appropriate behaviors and learning for students with diverse learning characteristics; and 4Q) analyzes student behavior data to develop and support positive behavior. Standard 5 - Instructional Delivery – The competent teacher differentiates instruction by using a variety of strategies that support critical and creative thinking, problem-solving, and continuous growth and learning. This teacher understands that the classroom is a dynamic environment requiring ongoing modification of instruction to enhance learning for each student. Knowledge Indicators – The competent teacher: 5A) understands the cognitive processes associated with various kinds of learning; 5B) understands principles and techniques, along with advantages and limitations, associated with a wide range of evidence-based instructional practices; 5C) knows how to implement effective differentiated instruction through the use of a wide variety of materials, technologies, and resources; 5D) understands disciplinary and interdisciplinary instructional approaches and how they relate to life and career experiences; 5E) knows techniques for modifying instructional methods, materials, and the environment to facilitate learning for students with diverse learning characteristics; 5F) knows strategies to maximize student attentiveness and engagement; 5G) knows how to evaluate and use student performance data to adjust instruction while teaching; and 5H) understands when and how to adapt or modify instruction based on outcome data, as well as student needs, goals, and responses. Performance Indicators – The competent teacher: 5I) uses multiple teaching strategies, including adjusted pacing and flexible grouping, to engage students in active learning opportunities that promote the development of critical and creative thinking, problem-solving, and performance capabilities; 5J) monitors and adjusts strategies in response to feedback from the student; 5K) varies his or her role in the instructional process as instructor, facilitator, coach, or audience in relation to the content and purposes of instruction and the needs of students; 5L) develops a variety of clear, accurate presentations and representations of concepts, using alternative explanations to assist students’ understanding and presenting diverse perspectives to encourage critical and creative thinking; 5M) uses strategies and techniques for facilitating meaningful inclusion of individuals with a range of abilities and experiences; 5N) uses technology to accomplish differentiated instructional objectives that enhance learning for each student; 5O) models and facilitates effective use of current and emerging digital tools to locate, analyze, evaluate, and use information resources to support research and learning; 5P) uses student data to adapt the curriculum and implement instructional strategies and materials according to the characteristics of each student; 5 5Q) uses effective co-planning and co-teaching techniques to deliver instruction to all students; 5R) maximizes instructional time (e.g., minimizes transitional time); and 5S) implements appropriate evidence-based instructional strategies. Standard 6 - Reading, Writing, and Oral Communication – The competent teacher has foundational knowledge of reading, writing, and oral communication within the content area and recognizes and addresses student reading, writing, and oral communication needs to facilitate the acquisition of content knowledge. Knowledge Indicators – The competent teacher: 6A) understands appropriate and varied instructional approaches used before, during, and after reading, including those that develop word knowledge, vocabulary, comprehension, fluency, and strategy use in the content areas; 6B) understands that the reading process involves the construction of meaning through the interactions of the reader's background knowledge and experiences, the information in the text, and the purpose of the reading situation; 6C) understands communication theory, language development, and the role of language in learning; 6D) understands writing processes and their importance to content learning; 6E) knows and models standard conventions of written and oral communications; 6F) recognizes the relationships among reading, writing, and oral communication and understands how to integrate these components to increase content learning; 6G) understands how to design, select, modify, and evaluate a wide range of materials for the content areas and the reading needs of the student; 6H) understands how to use a variety of formal and informal assessments to recognize and address the reading, writing, and oral communication needs of each student; and 6I) knows appropriate and varied instructional approaches, including those that develop word knowledge, vocabulary, comprehension, fluency, and strategy use in the content areas. Performance Indicators – The competent teacher: 6J) selects, modifies, and uses a wide range of printed, visual, or auditory materials, and online resources appropriate to the content areas and the reading needs and levels of each student (including ELLs, and struggling and advanced readers); 6K) uses assessment data, student work samples, and observations from continuous monitoring of student progress to plan and evaluate effective content area reading, writing, and oral communication instruction; 6L) facilitates the use of appropriate word identification and vocabulary strategies to develop each student’s understanding of content; 6M) teaches fluency strategies to facilitate comprehension of content; 6N) uses modeling, explanation, practice, and feedback to teach students to monitor and apply comprehension strategies independently, appropriate to the content learning; 6O) teaches students to analyze, evaluate, synthesize, and summarize information in single texts and across multiple texts, including electronic resources; 6P) teaches students to develop written text appropriate to the content areas that utilizes organization (e.g., compare/contrast, problem/solution), focus, elaboration, word choice, and standard conventions (e.g., punctuation, grammar); 6Q) integrates reading, writing, and oral communication to engage students in content learning; 6R) works with other teachers and support personnel to design, adjust, and modify instruction to meet students’ reading, writing, and oral communication needs; and 6S) stimulates discussion in the content areas for varied instructional and conversational purposes. Standard 7 - Assessment – The competent teacher understands and uses appropriate formative and summative assessments for determining student needs, monitoring student progress, measuring student 6 growth, and evaluating student outcomes. The teacher makes decisions driven by data about curricular and instructional effectiveness and adjusts practices to meet the needs of each student. Knowledge Indicators – The competent teacher: 7A) understands the purposes, characteristics, and limitations of different types of assessments, including standardized assessments, universal screening, curriculum-based assessment, and progress monitoring tools; 7B) understands that assessment is a means of evaluating how students learn and what they know and are able to do in order to meet the Illinois Learning Standards; 7C) understands measurement theory and assessment-related issues, such as validity, reliability, bias, and appropriate and accurate scoring; 7D) understands current terminology and procedures necessary for the appropriate analysis and interpretation of assessment data; 7E) understands how to select, construct, and use assessment strategies and instruments for diagnosis and evaluation of learning and instruction; 7F) knows research-based assessment strategies appropriate for each student; 7G) understands how to make data-driven decisions using assessment results to adjust practices to meet the needs of each student; 7H) knows legal provisions, rules, and guidelines regarding assessment and assessment accommodations for all student populations; and 7I) knows assessment and progress monitoring techniques to assess the effectiveness of instruction for each student. Performance Indicators – The competent teacher: 7J) uses assessment results to determine student performance levels, identify learning targets, select appropriate research-based instructional strategies, and implement instruction to enhance learning outcomes; 7K) appropriately uses a variety of formal and informal assessments to evaluate the understanding, progress, and performance of an individual student and the class as a whole; 7L) involves students in self-assessment activities to help them become aware of their strengths and needs and encourages them to establish goals for learning; 7M) maintains useful and accurate records of student work and performance; 7N) accurately interprets and clearly communicates aggregate student performance data to students, parents or guardians, colleagues, and the community in a manner that complies with the requirements of the Illinois School Student Records Act [105 ILCS 10], 23 Ill. Adm. Code 375 (Student Records), the Family Educational Rights and Privacy Act (FERPA) (20 USC 1232g) and its implementing regulations (34 CFR 99; December 9, 2008); 7O) effectively uses appropriate technologies to conduct assessments, monitor performance, and assess student progress; 7P) collaborates with families and other professionals involved in the assessment of each student; 7Q) uses various types of assessment procedures appropriately, including making accommodations for individual students in specific contexts; and 7R) uses assessment strategies and devices that are nondiscriminatory, and take into consideration the impact of disabilities, methods of communication, cultural background, and primary language on measuring knowledge and performance of students. Standard 8 - Collaborative Relationships – The competent teacher builds and maintains collaborative relationships to foster cognitive, linguistic, physical, and social and emotional development. This teacher works as a team member with professional colleagues, students, parents or guardians, and community members. Knowledge Indicators – The competent teacher: 8A) understands schools as organizations within the larger community context; 7 8B) understands the collaborative process and the skills necessary to initiate and carry out that process; 8C) collaborates with others in the use of data to design and implement effective school interventions that benefit all students; 8D) understands the benefits, barriers, and techniques involved in parent and family collaborations; 8E) understands school- and work-based learning environments and the need for collaboration with all organizations (e.g., businesses, community agencies, nonprofit organizations) to enhance student learning; 8F) understands the importance of participating on collaborative and problem-solving teams to create effective academic and behavioral interventions for all students; 8G) understands the various models of co-teaching and the procedures for implementing them across the curriculum; 8H) understands concerns of families of students with disabilities and knows appropriate strategies to collaborate with students and their families in addressing these concerns; and 8I) understands the roles and the importance of including students with disabilities, as appropriate, and all team members in planning individualized education programs (i.e, IEP, IFSP, Section 504 plan) for students with disabilities. Performance Indicators – The competent teacher: 8J) works with all school personnel (e.g., support staff, teachers, paraprofessionals) to develop learning climates for the school that encourage unity, support a sense of shared purpose, show trust in one another, and value individuals; 8K) participates in collaborative decision-making and problem-solving with colleagues and other professionals to achieve success for all students; 8L) initiates collaboration with others to create opportunities that enhance student learning; 8M) uses digital tools and resources to promote collaborative interactions; 8N) uses effective co-planning and co-teaching techniques to deliver instruction to each student; 8O) collaborates with school personnel in the implementation of appropriate assessment and instruction for designated students; 8P) develops professional relationships with parents and guardians that result in fair and equitable treatment of each student to support growth and learning; 8Q) establishes respectful and productive relationships with parents or guardians and seeks to develop cooperative partnerships to promote student learning and well-being; 8R) uses conflict resolution skills to enhance the effectiveness of collaboration and teamwork; 8S) participates in the design and implementation of individualized instruction for students with special needs (i.e., IEPs, IFSP, transition plans, Section 504 plans), ELLs, and students who are gifted; and 8T) identifies and utilizes community resources to enhance student learning and to provide opportunities for students to explore career opportunities. Standard 9 - Professionalism, Leadership, and Advocacy – The competent teacher is an ethical and reflective practitioner who exhibits professionalism; provides leadership in the learning community; and advocates for students, parents or guardians, and the profession. Knowledge Indicators – The competent teacher: 9A) evaluates best practices and research-based materials against benchmarks within the disciplines; 9B) knows laws and rules (e.g., mandatory reporting, sexual misconduct, corporal punishment) as a foundation for the fair and just treatment of all students and their families in the classroom and school; 9C) understands emergency response procedures as required under the School Safety Drill Act [105 ILCS 128/1], including school safety and crisis intervention protocol, initial response 8 actions (e.g., whether to stay in or evacuate a building), and first response to medical emergencies (e.g., first aid and life-saving techniques); 9D) identifies paths for continuous professional growth and improvement, including the design of a professional growth plan; 9E) is cognizant of his or her emerging and developed leadership skills and the applicability of those skills within a variety of learning communities; 9F) understands the roles of an advocate, the process of advocacy, and its place in combating or promoting certain school district practices affecting students; 9G) understands local and global societal issues and responsibilities in an evolving digital culture; and 9H) understands the importance of modeling appropriate dispositions in the classroom. Performance Indicators – The competent teacher: 9I) models professional behavior that reflects honesty, integrity, personal responsibility, confidentiality, altruism and respect; 9J) maintains accurate records, manages data effectively, and protects the confidentiality of information pertaining to each student and family; 9K) reflects on professional practice and resulting outcomes; engages in self-assessment; and adjusts practices to improve student performance, school goals, and professional growth; 9L) communicates with families, responds to concerns, and contributes to enhanced family participation in student education; 9M) communicates relevant information and ideas effectively to students, parents or guardians, and peers, using a variety of technology and digital-age media and formats; 9N) collaborates with other teachers, students, parents or guardians, specialists, administrators, and community partners to enhance students’ learning and school improvement; 9O) participates in professional development, professional organizations, and learning communities, and engages in peer coaching and mentoring activities to enhance personal growth and development; 9P) uses leadership skills that contribute to individual and collegial growth and development, school improvement, and the advancement of knowledge in the teaching profession; 9Q) proactively serves all students and their families with equity and honor and advocates on their behalf, ensuring the learning and well-being of each child in the classroom; 9R) is aware of and complies with the mandatory reporter provisions of Section 4 of the Abused and Neglected Child Reporting Act [325 ILCS 5/4]; 9S) models digital etiquette and responsible social actions in the use of digital technology; and 9T) models and teaches safe, legal, and ethical use of digital information and technology, including respect for copyright, intellectual property, and the appropriate documentation of sources.Problem Solving with Equations and Graphing and Writing Inequalities Practice Yaama I'm Jack Evans and you're watching BTN. Here's what's coming up. We uncover the story behind this famous photo, learn about First Nations seasons and find out the history of Book Week. What is Statehood? Reporter: Tatenda Chibika INTRO: But first, the Prime Minister Anthony Albanese has announced that Australia will join other countries in recognising Palestine as an independent state. So, what does that mean? Tatenda found out. Anthony Albanese, Prime Minister: Australia will recognise the state of Palestine. Australia will recognise the right of the Palestinian people to a state of their own. We will work with the international community to make this right a reality. Tatenda Chibika, Reporter: That's the moment our Prime Minister said Australia would recognise Palestine as an independent state at the upcoming United Nations General assembly next month. It's something other countries, including France and Canada, have said they'll be doing too. So, what does that mean exactly? To be considered an independent state under international law a place needs to have its own land or territories with defined borders, it needs to have people who permanently live there, have a working government and it has to be able to talk and make deals with other countries. Once a place meets all those rules, it can ask to be recognised by other independent states and countries. But a big step in becoming an independent state is being fully recognised by the United Nations. To do that you first need to get approval from at least nine members of the UN's Security Council. That's a group of countries responsible for maintaining international peace and security. But even then, that tick of approval can still be blocked by one of the Security Council's five permanent members Russia, China, the UK, the US and France. If the Security Council approves, the decision then goes to the UN's General Assembly where at least two thirds of the UN's 193 members have to agree to make it official. Yeah, it's a pretty complex process which is why we've only seen a handful of countries recognised by the UN in recent years like South Sudan and Montenegro. Others like Kosovo are only 'partially' recognised which means they have some recognition but not enough to become a full member state at the UN. Right now, Palestine is recognised by more than 140 countries — that's more than two thirds of the UN General Assembly. So, why hasn't it become a UN member state yet? Well, it came pretty close last year when 12 members of the Security Council voted in favour of it. VANESSA FRAZIER, AMBASSADOR OF MALTA, APRIL 2024 UNSC PRESIDENT: I shall now put the draft resolution to the vote. But the US, a close ally to Israel, used its special powers to block Palestine from becoming a member state. VANESSA FRAZIER: Those against? At the time, the U.S said Palestine and Israel needed to come to an agreement on their own first. Throughout the years, there have been attempts to figure out a way for both Palestine and Israel to exist peacefully alongside each other but that hasn't happened yet. And now Israel has said that recognising Palestine as an independent state would be rewarding Hamas the group in charge of Gaza which was responsible for the terror attacks on October 7th, 2023. But the Palestinian Authority which governs parts of the West Bank says Hamas won't have a role in any future state of Palestine which will exist peacefully alongside Israel. Australia, like the US, had previously said that it wanted Israel and Palestine to figure out things by themselves first but because of how the war has been going the Australian government is worried that if it continues to wait, there might not be a Palestinian state to recognise. ANTHONY ALBANESE, PRIME MINISTER: There has been too many lives lost, both Israeli's and Palestinians and the world is saying we need a solution to this conflict, we need to end the cycle of violence and the way to do that is to have a two-state solution. News Quiz Russia's President Vladimir Putin stepped foot on American Soil for the first time in a decade to meet with US President Donald Trump. What state did they meet in? Alabama, Alaska or Arizona?It's Alaska. The two leaders met to discuss a way to end the war in Ukraine but weren't able to make any final agreements. DONALD TRUMP, US PRESIDENT: There were many, many points that we agreed on. Most of them, I would say, a couple of big ones, that we haven't quite got there, but we've made some headway. There's no deal until there's a deal. A lot of people criticised the two world leaders for not including Ukraine's president Volodymyr Zelenskyy in the meeting. But that didn't seem to worry Mr Trump who said the meeting was a success and Mr Putin even invited the US President to meet up again in Russia. DONALD TRUMP: We'll see you again very soon. Thank you very much, Vladimir. VLADIMIR PUTIN, RUSSIAN PRESIDENT: Next time in Moscow. DONALD TRUMP: Oh, that's an interesting one. No, no, no. I'll get a little heat on that one. Last week thousands of people marked the 80th anniversary of VJ Day. What does VJ Day commemorate? The victory of Allied forces in Europe, the surrender of Japan and the end of World War II or the dropping of the first atomic bomb? VJ Day or Victory over Japan day commemorates the surrender of Japan and the end of World War II on the 15th of August 1945. Around the world, and here in Australia, people marked the anniversary with ceremonies remembering those who fought in the war. REPORTER: Who will you be remembering today? VETERAN: Oh, a lot of fellows that I knew that never made it home. Scientists in the UK have created toothpaste that includes which of these ingredients? Hair, eye lashes or fingernails? Yeah, they're all a bit random and gross but the answer is hair. According to scientists from King's College in London, hair could be the key to good oral health because it contains a protein called Keratin which they say when mixed with saliva forms a crystal-like protective coating similar to enamel. And Swifties rejoice because Taylor Swift has announced her 12th Studio album. It's called life of a show what? Is it show pony, show girl or show bag? It's Life of a Showgirl and it'll be released October 3rd. Vincent Lingiari Reporter: Joseph Baronio INTRO: Now to this very famous photograph. It was taken 50 years ago and depicts a really significant moment in Australian history. Joe found out about the story behind it. On the 16th of August 1975, this famous photo was taken. It shows the former Prime Minister Gough Whitlam pouring sand into the hand of Aboriginal leader Vincent Lingiari. A simple gesture that symbolised handing the land at Wave Hill in the Northern Territory back to the Gurindji people. But the journey to get there was far from simple. It started back in the 1960s. At the time, Wave Hill was the biggest cattle station in the world, controlled by British landowner Lord Vestey. The Gurindji people, who had lived on the land for generations, worked for Vestey, but they weren't paid fairly, and conditions were tough. NEWS REPORTER: The station's 100 aboriginal stockmen, with their 100 dependents, are camped in the dry bed of the Victoria River with little shade from 90-degree heat, dust and flies. Eventually, Gurindji leader Vincent Lingiari said it was time to act. VINCENT LINGIARI: I said, "What was it before Lord Vestey born and I was born?" It was blackfella country. So, on August 23rd, 1966, Mr Lingiari and his fellow Aboriginal workers went on strike. It became known as the Wave Hill Walk Off. They moved their camp away from the Wave Hill station to a sacred site called Daguragu on Wattie Creek. They wanted to set up their own cattle station, and said they wouldn't move until their land was returned to them. For years, petitions and negotiations went on between the Gurindji people, the NT Administration, and the Australian Government in Canberra. CLAPPERS: 31. 32. 33. DAVID QUINN, ABSCOL: Well, it's basic justice that their land is recognised. PROTESTORS: Equal rights! As the news spread across the country, thousands of Aussies joined the campaign, including the leader of the Labor Party, Gough Whitlam, who made this promise during his 1972 election campaign. GOUGH WHITLAM: We will legislate to give Aborigines land rights. Not just because their case is beyond argument, but because all of us as Australians are diminished, while the Aborigines are denied their rightful place in this nation. Later that year, Gough Whitlam became Prime Minister. (Song From Little Things Big Things Grow, Song by Kev Carmody and Paul Kelly, 1993) From little things big things grow,from little things big things grow… But it wasn't until 1975, 9 years after the Wave Hill Walk Off started, that he followed through with his promise. Eight years went by, eight long years of waiting'Til one day a tall stranger appeared in the landAnd he came with lawyers and he came with great ceremony GOUGH WHITLAM: I solemnly hand to you these deeds as proof in Australian law that these lands belong to the Gurindji people. And through Vincent's fingers poured a handful of sandFrom little things big things grow 50 years on, and The Wave Hill Walk Off is seen as a pivotal moment in Australia's history. It led to significant legal and social changes for First Nations people, which is something many agree is worth celebrating. First Nations Seasons Reporter: Saskia Mortarotti INTRO: Recently, Melbourne's Lord Mayor suggested ditching the four-season calendar that most of us are familiar with and adopting a six-season Wurundjeri calendar instead saying it gives a better description of what the weather's actually like there. Sas found out more about the different seasonal calendars used by First Nations people. SASKIA MORTAROTTI, REPORTER: Right now, in most of the country, it's pretty cold. COLD GIRL: Think of somewhere warm. What? It's 32 degrees in Darwin in the middle of winter? But ah, yeah. There are some places where it's, well, quite warm. Which makes you wonder whether the weather actually matches the seasons. You see, Australia is pretty big, and we have lots of different weather patterns. Which is something First Nations people have tracked for thousands of years with their own seasonal calendars. KARL WINDA TELFER, CULTURAL CREATIVE KANYANYAPILLA: Why have we got four seasons when you know that don't make any sense here. It doesn't relate to the country here. This is Karl Telfer. He's an artist and storyteller who produced the Kuri Kurru exhibition at the Museum of Discovery in Adelaide that explores the 6 different seasons of the Kaurna Meyunna. SASKIA MORTAROTTI: So, how do you know when you're in one of those six seasons? KARL WINDA TELFER: Well, there are stars that rise. So, you know, there are certain stars, like in Parnatti, for example. There's a star called Parna, and we know what that star is. So, that talks to us about, okay, the time now is going to be cold on the ground. First Nations calendars like the Kaurna one don't just tell us what's happening with the weather; they're also used to track when certain plants and animals are around. KARL WINDA TELFER: It teaches you about what plants you can, you know, what you can eat what you can't and all that what is ready certain times a year and fruit everything, bird shows you the right time to eat the fruit, perfect time, if you try and go get them the next week they're gone. Karl says we can also use these calendars to see how the environment has changed over time. KARL WINDA TELFER: Kudlilla is the season we're in now and Kudlilla that talks about like the rain but we're not having enough rain these days, well, these times. And this is due to climate and the climate changing. There are many different First Nations seasonal calendars around the country. Like Ngan'gi calendar from the Northern Territory which has 13 seasons that follow the life cycle of the native spear grass. Or the Wurundjeri Calendar in Victoria which has 6 seasons. And recently, Melbourne's Lord Mayor, Nicholas Reece, said Melbourne, or Naarm, would be better off adopting the Wurundjeri calendar because it's more in tune to what's happening with the weather. Something many, including Karl, think we should be doing right across the country. KARL WINDA TELFER: I'm talking about the English four seasons. So, this is totally different systems that we're talking about and weather patterns and currents and all sorts of different things, because it's the sea country too. So, my question is, well, why do we have that? If that doesn't work, you know? Quiz How many seasons are there in the Tiwi Island Calendar? 1, 2 or 3? It's 3, although they also have 13 minor seasons. Book Week Reporter: Wren Gillett INTRO: This week, kids across Australia have been dressing up as their favourite characters to celebrate Book Week. Wren finds out why Book Week began 80 years ago and why it's still important today for getting young Aussies into reading. STUDENT: I read an hour every night, maybe even two hours some nights. STUDENT: My favourite book series are the Harry Potter series and the Keeper of the Lost City series. STUDENT: Probably Bad Guys and Weirdo. STUDENT: I like the Amulet, I've been reading that. STUDENT: I love reading Dork Diaries and Exploding Endings. Whether it's Fantasy, mystery, history — whatever you're into. Book week is a time to celebrate, well, books. STUDENT: Me and my friends are dressing up as Inside Out. STUDENT: I was thinking SpongeBob. STUDENT: I'm dressing up as Winnie the Pooh and it's just a fun way to express what kind of books you like. And guess what, book week has actually been a thing for many, many years. WREN GILLETT, REPORTER: Once upon a time, in a land not so far away, literacy lovers noticed a problem. The year was 1945. The second World War had just ended, and kids were mainly reading books from overseas, in particular the UK. Because, at the time, there weren't many Aussie authors writing books for children. WREN GILLETT: So, a group of passionate teachers, librarians, booksellers, publishers, and book-loving volunteers, decided to create what we now know as The Children's Book Council of Australia. Familiar logo, right? Together, they launched book week, all in an effort to get Aussie kids' reading more. And it seemed to work. The 1960s saw a boom in Australian children's books being published. REPORTER: How many books do you read a week? STUDENT: Well, it really depends on the week. If there's exams, I might read only one or two. But if there's no exams and if I've got plenty of time, I might read up to five or six. WREN GILLETT: But today, it's a slightly different story. Studies show that less than one in five eight to 18-year-olds are reading in their free time, and that only one in three actually enjoy reading for fun. WREN GILLETT: Why do you reckon we're seeing this trend? STUDENT: People are getting sucked into screens and they're like spending hours just scrolling through TikTok and stuff, and they're getting so attached to it that they don't feel the need to pick up books and read them. Yeah, there's a lot of different things competing for our attention these days, but many think books are still worth our time. PETER HELLIER, AUSSIE COMEDIAN AND AUTHOR: Books are the exact opposite of boring. And if you think they're boring, I'm sorry, but you're wrong. This is Peter Hellier, he's a pretty famous Aussie comedian, actor, and the author behind these books. And he's just released another one called Detective Galileo, about a trail horse who dreams of solving crimes. PETER HELLIER: He joins the police force and quickly finds out that the horses don't actually solve the crimes, it's the police officers who solve the crime. So he promptly gets thrown out of the force and begins his own detective agency, which I'm reliably told is the only detective agency in the world run by a horse. Peter actually started writing books when he was a kid. PETER HELLIER: I started writing when I was six, seven, eight years old. In fact, I started my own publishing company called Better Books. And I would write these books, and then I would get a parent or one of my parents or teachers to type them up. And I would read them in front of the class. And, you see, each has the logo, the Better Books logo, there it is — the famous Better Books logo. WREN GILLETT: You weren't mucking around. PETER HELLIER: There all on all of them. There we go. There we go. Many, Including Peter, say there's plenty to get from a good book. They help us learn new words and phrases, get a better understanding of the world around us, and strengthen our imaginations. PETER HELLIER: Books can take you absolutely anywhere. They can take you to countries that you never dreamed about going. Countries that exist, countries that don't exist. Reading just makes the world a much bigger place. It's why for the past 80 years, schools around the country have been taking part in book week. STUDENT: Reading is a place where you can have your own world just to yourself. STUDENT: It's like watching a movie inside your head, but you can choose how it goes. STUDENT: Picking up a book is a good idea, maybe you should start with something that you're interested with and then you can start exploring from there. Quiz What is the title of the book that took out this year's Book of the year Award for younger readers? It's Laughter is the Best Endingby Maryam Master. Some other winners included I'm not really here by Gary Loneborough which took out book of the year for older readers and best picture book went to The Truck Cat, by Deborah Frenkel. Sport Australia's men's national basketball team — the Boomers — have won their third Asia Cup in a row, with an epically narrow victory over China. COMMENTATOR: It is Australia who are celebrating! China started strong, leading 25-17 at quarter time. But Aussie Xavier Cooks led a fierce comeback, shooting 30 points and collecting nine rebounds, earning him the title of MVP. And there seriously couldn't have been a tighter finish. Just as the final buzzer went off, China missed a shot that would have won them the game, leaving Australia with a 90-89 victory. COMMENTATOR: An unbelievable finish. The 2025 AFLW season kicked off last week, and so did a new trick. Yeah, 19-year-old Ash Centra from Collingwood, pulled out this move in the warm-up before their season-opener to Carlton, and since then, a lot of people have been trying to do it, with some success, kind of? FOOTY PLAYER: No, I'm not doing it on camera. But despite the epic warmup, Carlton did end up beating Collingwood by 24 points. Now, the moves from these athletes in China weren't quite so graceful but give 'em a break, okay, they're robots. For the first time ever, humanoid robots from all over the world, competed in their very own games, which featured, soccer, boxing, running, and ahh, lots of falling over. Lots. Luckily though, they did bring their own cheer squad. Young Rapper Reporter: Rylie INTRO: Finally, we're going to meet another winner of this year's Heywire competition — which asks young people in regional areas to share their stories. Rylie's going to tell us how music helped to transform his life. Check it out. Mum and I were homeless. We lived at a caravan park, in motels and tents around Warrnambool. It wasn't pretty. I'm First Nations, and I remember feeling like, my own country is failing me right now. So, we camped right along here. I remember pitching a tent right here and this was actually around the same time I started to get into music which was a good way for me to have something to look forward to. I was raised by the SoundCloud era, listening to a lot of trap music. When I was in school, I'd rap along to songs by Juice World, then I started to make my own. My first track was recorded on my phone. It was bad but a lot of fun to make. Some kids in my school heard it and shamed me. That put me off music for the next couple of years, until a friend of mine bought a microphone and encouraged me to give it another go. There was something about that mic and the energy of the crew around me that gave me confidence. It lit a fire in me. Over time, I was able to focus my flow. My songs are about escapism, living the life, being a success. I rap about stuff that takes me to a better place in my head. I'm manifesting my future. My stage name is Hundo Milli, it's short for hundreds of millions. Money's not really the end goal; it's more about having the freedom to dream big. Mum taught me to never stop believing. Even when times were tough, she kept pushing for us to get housing and eventually we did. We're some of the lucky ones. Today, I'm in a Melbourne studio recording my next single. I remember living in my tent dreaming about this very moment and now I'm here, doing what I love. Ain't nothing going to stop me. Closer Well, that's all we've got for you today, but we'll be back before you know it. In the meantime, you can head to our website, there's plenty to see and do. You can also catch Newsbreak every weeknight and there's BTN High for all you highschoolers out there. Have an awesome week and I'll see you next time. Bye.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 •(-9.8m/s2) •d 0 m2/s2 = 686.44 m2/s2 + (-19.6 m/s2) •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.