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Final Consonant Blends
Quiz by Jessica Norton
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Which word has the final blend "-st"?
Choose the word with the final blend "-nd".
Which word has the final blend "-st"?
Choose the word with the final blend "-nd".
Which word ends with the blend "-nt"?
Select the word with the final blend "-nk".
Which word has the final blend "-mp"?
Choose the word with the final blend "-ng".
Which word ends with the blend "-st"?
Select the word with the final blend "-nd".
Choose the word with the final blend "-nt".
Which word has the final blend "-nk"?
Final Stable Syllables and Consonant-le - Starter QuizCan you create an evaluation using this information PHONETICS VS. PHONOLOGY Whereas phonetics is the study of sounds that occur in language, phonology is the study of how these sounds are organized and how they function in language. It uses the classifications of sounds derived from phonetics to describe and analyze how sounds occur in speech. STRUCTURALIST PHONEMICS STRUCTURALIST PHONEMICS As linguists began to study sounds in fine detail, they recognized increasingly complex aspects of phonetic organization. For example, the sound /p/ appears in different varieties in English. STRUCTURALIST PHONEMICS One of the varieties of /p/ is indicated by [ph]. This sound is produced with an accompanying puff of air called aspiration, as in the words “pill,” and “peace.” Another sound, indicated by [p•], is produced when there is little or no aspiration; this sound occurs in a word like “spill.” A third major variety for the /p/ sound is the unreleased [p– ], which may occur at the end of a word like “stop.” To deal with these variations for the /p/ sound, the structuralists suggested the existence of an abstract unit which they termed a phoneme. STRUCTURALIST PHONEMICS A phoneme was defined by the structuralists as an abstract phonological unit that represents a class of real sounds, termed the allophones of a phoneme. The phoneme /p/ in English, then, is represented by the allophones [ph], [p•], and [p– ]. STRUCTURALISTS: MINIMAL PAIRS How do we know what these abstract units of sound called phonemes are? In order to find the phonemes of a language, the structuralists developed the concept of the minimal pair, defined as any two words that: a) Contain the same number of segments b) Differ in meaning c) Exhibit only one phonetic difference. STRUCTURALISTS: MINIMAL PAIRS In practical terms, phonemes distinguish meanings; and a phoneme can also be defined as the smallest meaning-distinguishing unit of sound. For instance, the words “pin” /pɪn/ and “bin” /bɪn/ mean different things, and the only one difference in these words occurs in the initial sounds. STRUCTURALISTS: MINIMAL PAIRS By using the concept of a minimal pair, we can determine that the three variations of the /p/ sound do not represent three phonemes. Certainly, it is possible to pronounce the word cap with either an aspirated [ph ] or unreleased [p– ]; however, the two forms [kæph ] and [kæp– ] are not a minimal pair, even though they involve different sounds, because they are identical in meaning. STRUCTURALISTS: FREE VARIATION The two forms [kæph ] and [kæp– ] are, therefore, said to exhibit free variation: that is, the pronunciation may vary without signifying a change in meaning. In other words, we may conclude that the unreleased [p– ] and the aspirated [ph ] are not representations of different phonemes in English; they are, in fact, allophones of one phoneme, /p/. STRUCTURALISTS: COMPLEMENTARY DISTRIBUTION When phonemes have more than one allophone in a language, the allophones are said to be in complementary distribution. Complementary distribution means that the allophones of a phoneme occur in different phonetic environments (that is, with different sounds surrounding them). TRANSFORMATIONAL- GENERATIVE PHONOLOGY TRANSFORMATIONAL-GENERATIVE PHONOLOGY Transformational-generative phonology is a relatively recent development in linguistic theory. Chomsky launched Transformational-Generative Grammar in 1957, but the earliest studies within this framework were largely concerned with syntax. A decade later, the first comprehensive transformational-generative treatment of English phonology appeared: Chomsky and Halle’s The Sound Pattern of English (1968). TRANSFORMATIONAL-GENERATIVE PHONOLOGY Transformational-generative phonologists strongly oppose the structuralists’ phonemic level. They replace this level by a series of rules that directly relate underlying representations to observed phonetic representations. The central mechanisms in transformational-generative phonology, then, are underlying representations and phonological rules. PHONOLOGICAL RULES A rule is an operational statement in which some linguistic entity is modified, resulting in a new linguistic entity. Rules may add elements, remove elements, or change elements. By using phonological rules, linguists attempt to demonstrate that there is order in linguistic phenomena and that linguistic patterns are systematic. PHONOLOGICAL DERIVATION A phonological derivation is an operation that begins with an underlying representation and, through the application of a set of specific rules, yields the actual sound the speaker produces. The representation of a phonological rule has the following general appearance. /A/ → [B] / C “A” changes to “B” under condition “C” PHONOLOGICAL RULE – EXAMPLE In most Southern dialects, the word ten is pronounced like the word tin. This is not an isolated fact, for den is pronounced like din and Ben is pronounced like bin, and so on. This very general fact can be represented by the phonological rule: /ɛ/ → [I] / ___ [n] den /dɛn/ → /dIn/ Ben /bɛn/ → /bIn/ ten /tɛn/ → /tIn/ /ɛ/ → [I] / ___ [n] - high - low - tense + front + high - tense + front + sonorant + anterior + coronal - continuant NOTATIONAL DEVICES IN PHONOLOGICAL RULES The statement of phonological rules can be complex, and linguists have developed several notational devices for writing them. Often, the following symbols will be necessary for stating the conditions under which rules apply: # indicates a word boundary + indicates an intraword boundary $ indicates a syllable boundary UNDERLYING REPRESENTATIONS AND RELATED ISSUES The transformational-generative description of phonology relates underlying representations to phonetic representations by rules. This can be represented in a simple example: In English, there are certain pairs of words like sign / signature, and malign / malignant that exhibit a regular alternation in their phonetic representations: [g] is present in the second member of the pairs but absent in the first member. UNDERLYING REPRESENTATIONS AND RELATED ISSUES To explain the relatedness of words such as sign / signature, we could claim that the underlying representation of the segment in all such pairs is /g/ and that a rule operates to delete /g/ before syllable-final nasals. Thus, the rule “/g/ is deleted before syllable-final nasal” would appear formally as: + voice - anterior →∅ ____ [+ nasal] $ - coronal UNDERLYING REPRESENTATIONS AND RELATED ISSUES On the left-hand side of the arrow, we place the features needed to uniquely specify /g/ among the consonants; that is, no other consonant has the features [+ voice], [- anterior], and [- coronal]. The symbols → mean that the sound /g/ changes to nothing or more properly “/g/ is deleted.” The horizontal line following the slash mark refers to the position of /g/ - namely, before a segment that is [+nasal]. Finally, this [+nasal] segment occurs before a syllable boundary, as indicated by $. A less formal way of writing this rule would be: /g/ → / _ [+nasal] $ Notice that this rule also helps describe such alternations as phlegm/phlegmatic and paradigm/paradigmatic. Application Activity: Think of other words in which this rule can be applied. Write the sound segments to prove /g/ is deleted. Another example is the process through which the prefix meaning “not” is added to words. This prefix alternates among the forms /Im/, /In/, and /Iŋ/, depending on the point of articulation of the initial segment of the following word. -If the segment begins in the extreme front part of the mouth (labials), the form is /Im/, as in improper. -If the segment begins in the extreme back part of the mouth (velars), the form is /Iŋ/, as in incomplete. -If the segment begins in the mid-region of the mouth (all other sounds), the form is /In/, as in indecent. *Exceptions:Words beginning with /r/ or /l/. Analyze the Word “in + complete,” for example. /n/ → [ŋ] / __ [k] - continuant - continuant - continuant + sonorant → + sonorant - sonorant + anterior - anterior - strident + coronal - coronal - coronal + tense THE VELAR SOFTENING RULE Still another example of alternation in English is found in pairs of words like “electric / electricity,” in which the segments /k/ and /s/ alternate. /k/ changes to [s] only before non- low, front vowels. THE VELAR SOFTENING RULE /k/ → [s] / __ - continuant + continuant - strident → - sonorant V - anterior + anterior - low - coronal + coronal - backCourse of your preparation:
Synonyms: Study plan, curriculum of readiness, training regimen
Example sentence: "Create a comprehensive course of your preparation to ensure success in the upcoming exams."
Urdu meaning: تیاری کا مسلسل منصوبہ
Most prestigious:
Synonyms: Highest esteemed, most honored, preeminent
Example sentence: "Winning the Nobel Prize is considered one of the most prestigious achievements in the academic world."
Urdu meaning: سب سے عظیم
Hallmark of prestige:
Synonyms: Symbol of honor, distinctive feature of esteem, badge of high regard
Example sentence: "The commitment to quality is the hallmark of prestige for this renowned institution."
Urdu meaning: عظمت کا نشان
Disposal:
Synonyms: Management, arrangement, handling
Example sentence: "The resources at your disposal should be utilized wisely to achieve optimal results."
Urdu meaning: دستیابی
Come out with flying colors:
Synonyms: Excel, succeed, triumph
Example sentence: "Despite the challenges, she came out with flying colors in her final exams."
Urdu meaning: کامیاب ہونا
Keep in mind:
Synonyms: Bear in mind, remember, be mindful of
Example sentence: "Keep in mind the importance of time management when working on your projects."
Urdu meaning: ذہن میں رکھیں
Strive:
Synonyms: Endeavor, work hard, exert
Example sentence: "Strive for excellence in all your endeavors, and success will follow."
Urdu meaning: محنت کرنا
Epitomize brilliance:
Synonyms: Symbolize excellence, represent brilliance, embody perfection
Example sentence: "Her groundbreaking research work epitomizes brilliance in the field of science."
Urdu meaning: عظمت کو مثال بنانا
Jump out:
Synonyms: Stand out, emerge, be noticeable
Example sentence: "His exceptional talent allowed him to jump out among the competitors."
Urdu meaning: نمایاں ہونا
Appropriate:
Synonyms: Suitable, fitting, proper
Example sentence: "Choose the appropriate tools for the task to ensure efficiency."
Urdu meaning: مناسب
Digestible format:
Synonyms: Easily understood layout, comprehensible structure, user-friendly arrangement
Example sentence: "Present the information in a digestible format for better comprehension by the audience."
Urdu meaning: قابل فہم فارمیٹ
Without waffle:
Synonyms: Concise, to the point, without unnecessary elaboration
Example sentence: "Provide a clear and concise report without waffle to convey the key findings."
Urdu meaning: بے بکوفی
Reiteration:
Synonyms: Repetition, reaffirmation, restatement
Example sentence: "The constant reiteration of the importance of teamwork reinforced the company's values."
Urdu meaning: تکرار
Stupendous speculations:
Synonyms: Remarkable conjectures, extraordinary hypotheses, astounding suppositions
Example sentence: "The scientist presented stupendous speculations that challenged existing theories."
Urdu meaning: حیرت انگیز تخیلات
Superfluous language:
Synonyms: Unnecessary wording, excessive language, redundant expression
Example sentence: "Avoid using superfluous language in your writing to maintain clarity and precision."
Urdu meaning: زیادہ سے زیادہ الفاظ
Individual tales:
Synonyms: Personal narratives, unique stories, individual anecdotes
Example sentence: "The book is a collection of individual tales that showcase the diversity of human experiences."
Urdu meaning: انفرادی قصے
Long Call Option Trading Strategy: Learn the Basics LONG CALL SUMMARY Purchasing a call option is a bullish strategy that gives the buyer the right, but not the obligation, to buy 100 shares of the underlying asset at a specified strike price on or before the expiration date. This strategy is typically employed when an investor believes that the price of the underlying asset will increase in the future. The value of a call option is influenced by several factors, including the underlying asset's price, the strike price, the time to expiration, and implied volatility. As the price of the underlying asset increases and approaches or breaches the long call's strike price, the option's value will appreciate. This is because the option holder has the right to buy the underlying asset at a lower price than the current market price, resulting in a potential profit. Out-of-the-money (OTM) calls have a strike price that is higher than the current market price of the underlying asset. These options are typically cheaper than in-the-money (ITM) calls, which have a strike price lower than the current market price. ITM calls have intrinsic value, which is the difference between the strike price and the current market price, and extrinsic value, which is the additional premium paid for the option's time value. Extrinsic value decays over time as the option approaches expiration, and this can cause the option to lose value, especially if the underlying asset does not move towards the strike price. LONG CALL OPTION Purchasing a call option grants you the privilege, but not the responsibility, to buy 100 shares of the underlying asset at the specified strike price on or before the expiration date. This option grants you the flexibility to capitalize on potential price increases of the underlying asset. The value of a call option is positively correlated with the price of the underlying asset. As the price of the stock or ETF rises and approaches your strike price, the value of your call option increases. This is because the difference between the market price and the strike price widens, giving you a greater potential profit. This characteristic makes call options suitable for bullish strategies where investors anticipate price increases. Conversely, the value of a call option diminishes when the price of the underlying asset drops or remains constant. Time decay, which refers to the gradual loss of an option's value as its expiration date approaches, also contributes to the depreciation of call options. Over time, the intrinsic value of the option, which represents the difference between the strike price and the underlying asset's market price, decreases as the option nears expiration. Additionally, if the price of the underlying asset remains below the strike price, the option may expire worthless, resulting in a total loss of the premium paid. Understanding these dynamics is crucial when trading call options. It allows you to make informed decisions about when to enter and exit positions, taking into account factors such as the underlying asset's price movements, time decay, and market sentiment. Buying call options can provide an alternative strategy to gain long exposure to a stock's price movement without the need for purchasing shares directly. This approach, known as a long call position, offers the potential advantage of lower capital outlay compared to buying shares outright. However, it's crucial to understand the concept of time decay, which significantly impacts the value of long call options. Time decay refers to the gradual decrease in the value of an option as time passes. This phenomenon occurs due to two primary factors: theta and vega. Theta measures the rate at which an option's value decays over time, while vega measures the sensitivity of an option's price to changes in implied volatility. As the expiration date of the call option approaches, both theta and vega work together to erode the option's value. Consequently, to offset the impact of time decay, the underlying stock price must rise at a greater velocity towards the call option's strike price. This is because the intrinsic value of a call option, which represents the difference between the strike price and the underlying stock's current market price, increases as the stock price moves higher. Another important consideration when evaluating call options is the distinction between out-of-the-money (OTM) and in-the-money (ITM) calls. OTM calls have a strike price higher than the current market price of the underlying stock, while ITM calls have a strike price lower than the current market price. OTM calls are typically less expensive than ITM calls because their value is composed entirely of extrinsic value. Extrinsic value refers to the portion of an option's price that is not attributable to its intrinsic value. ITM calls, on the other hand, have both intrinsic and extrinsic value, resulting in a higher cost per contract. As time relentlessly marches forward, the value of call options undergoes a transformation. The extrinsic value, which represents the premium paid for the potential of future price movements, steadily diminishes as expiration approaches. This decay is universal, affecting all call options regardless of their initial strike price or distance from the underlying asset's current price. However, amidst this gradual erosion of extrinsic value, ITM (in-the-money) call options stand as an exception. These options retain their intrinsic value at expiration, which is the difference between the strike price and the underlying asset's price. This characteristic sets ITM call options apart from their OTM (out-of-the-money) counterparts, whose extrinsic value decays entirely to zero near or at expiration. The distinction between ITM and OTM call options underscores the significance of carefully considering both the time frame and strike price when making investment decisions. Traders seeking to maximize their potential gains through call options must be mindful of the impending decay of extrinsic value as expiration draws near. For long ITM call options, the ideal scenario is for the underlying asset to exhibit a significant upward movement. Such a price increase would enhance the intrinsic value of the option, making it worth more at expiration than the initial purchase price. This scenario holds true for OTM call options as well, as they require the underlying asset to move ITM at expiration to possess any value. Prior to expiration, both OTM and ITM call options have the potential to gain a combination of extrinsic and intrinsic value if the stock exhibits a rapid upward trajectory. This dynamic underscores the importance of monitoring market conditions and adjusting investment strategies accordingly. Understanding the Interplay of Time, Strike Price, and Option Value in Call Option Trading: In the realm of call option trading, comprehending the intricate interplay between time, strike price, and option value is paramount to success. These three factors collectively shape the dynamics of call option contracts, allowing traders to make informed decisions and capitalize on market opportunities. Time (Days to Expiration): Time, measured in days until expiration, is a crucial element in call option trading. As expiration approaches, the value of a call option is directly influenced by the time premium. The closer an option gets to expiration, the less time value it holds. This time decay accelerates in the final days leading up to expiration. Therefore, traders must carefully consider the time factor when selecting their expiration dates. Strike Price: The strike price represents the predetermined price at which the underlying asset can be bought (in the case of a call option) or sold (in the case of a put option). When choosing a strike price, traders must assess the current market price of the underlying asset and make an educated guess about its future direction. ITM (In-the-Money) call options are those with a strike price below the current market price, while OTM (Out-of-the-Money) call options have a strike price above the current market price. Option Value: Option value refers to the premium paid by the buyer of an option contract to the seller. This premium comprises two components: intrinsic value and time value. Intrinsic value is the difference between the strike price and the underlying asset's current market price. Time value, as mentioned earlier, is the premium paid for the remaining time until expiration. Auto-Exercise and Expiration Scenarios: Auto-Exercise: Long call options that expire ITM by $0.01 or more will be automatically exercised. This means that the buyer of the call option has the right to purchase the underlying asset at the strike price. If the investor holds only a long call, this will result in 100 long shares per contract purchased at the call option's strike price. On the other hand, investors holding the corresponding short shares will cover or buy shares at the call option's strike price. Expiration Worthless: Any long call options that expire OTM will expire worthless. In this scenario, the investor loses the entire premium paid for the contract, resulting in a maximum loss. Understanding these concepts is instrumental in developing effective call option trading strategies. By carefully considering the interplay between time, strike price, and option value, traders can position themselves to make profitable trades and minimize potential losses. PROFIT & LOSS DIAGRAM OF A LONG OTM CALL A long OTM call option can be profitable if the current market value of the option exceeds the price paid to purchase it. This can occur in two main scenarios: Stock Price Surpasses Strike Price: If the underlying asset's price rises above the strike price of the call option by more than the premium paid for the option, the call option becomes profitable. This is because the intrinsic value of the call option (the difference between the strike price and the underlying asset's price) becomes positive, and the call option can be exercised to purchase the underlying asset at a price below the market price. OTM Call Moves Closer to Underlying Asset Price: Even if the underlying asset's price does not reach the strike price, a long OTM call can still be profitable if the option's price increases. This can happen when there is a quick rally in the underlying asset's price, causing the call option's price to increase as well, even if the strike price is not reached. This is because the time value of the call option increases as the expiration date approaches, and the call option becomes more likely to be in the money. However, it's important to note that long OTM call options can also result in losses if the underlying asset's price does not surpass the breakeven point. The breakeven point is the price at which the call option's intrinsic value becomes equal to the purchase price of the option. If the underlying asset's price remains below the breakeven point until expiration, the call option will expire worthless, and the investor will lose the entire amount paid for the option. The maximum profit potential of a long OTM call option indeed has no theoretical limit, as a stock's price can theoretically rise indefinitely. This means that if the underlying stock price increases significantly, the call option holder can potentially reap substantial profits by exercising the option and buying the stock at the predetermined strike price. On the downside, the maximum loss on a long call option is limited to the premium paid for the option. This premium represents the total amount invested in the option contract and acts as a protective barrier against further losses. If the stock price declines or stays below the strike price at expiration, the option will expire worthless, and the investor will lose the entire premium paid. The flattened red loss zone in the diagram illustrates this limited loss potential. This zone represents the range of stock prices below the strike price at expiration where the option holder will lose money. The loss amount decreases as the stock price approaches the strike price and becomes zero when the stock price equals the strike price. Beyond the strike price, the option holder starts to make a profit. It's important to note that while the maximum profit potential is theoretically unlimited, it is highly unlikely for a stock price to rise dramatically within the short timeframe of an OTM option's expiration period. Therefore, while the potential rewards can be significant, the probability of achieving them is relatively low. PROFIT & LOSS DIAGRAM OF A LONG ITM CALL ITM (In-the-Money) options have a unique characteristic where the price of their intrinsic value directly correlates with the underlying asset's price. This means that for every one point movement in the underlying asset's price, the ITM option's intrinsic value moves by the same amount. While purchasing an ITM option provides immediate intrinsic value, it does not guarantee profitability upon execution. Similar to buying an OTM (Out-of-the-Money) call option, the purchase price of an ITM call must increase for it to be profitable. This requires the stock price to move further above the call strike price. This relationship is visually represented in the diagram, where the red and green zones converge on the x-axis. The maximum potential loss on a long call option is limited to the debit paid for the option, which is represented by the flattened red area in the diagram. This means that the most an investor can lose on a long call is the premium paid for the option, regardless of how far the underlying asset's price moves below the strike price. Understanding the price dynamics and potential risks associated with ITM options is crucial for traders and investors. While ITM options offer immediate intrinsic value, careful analysis and consideration of market conditions are necessary to determine their potential profitability. EXAMPLE OF A LONG OTM CALL OPTION XYZ currently trading @ $45 Buy to Open +1 XYZ 50-strike call @ $4 debit Cost: $4 debit ($400 total, ($4 x 100 shares)) Time Decay Affect Works against the option’s value Max Profit Theoretically unlimited Max Loss Debit paid per contract ($400) Breakeven Price (at expiration) Strike price + debit paid ($54) Account Type Required Cash, Margin, and IRA EXAMPLE OF A LONG ITM CALL OPTION XYZ currently trading @ $45 Buy to Open +1 XYZ 40-strike call @ $7 debit ($5 intrinsic value + $2 extrinsic value) Cost: $7 debit ($700 total) Time Decay Affect Works against the option’s value Max Profit Theoretically unlimited Max Loss Debit paid per contract ($700) Breakeven Price (at expiration) Strike price + debit paid ($47) Account Type Required Cash, Margin, and IRAIntroduction 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.final110.31.b.17.C Topic: Reading/Vocabulary Development