Position: The location of an object; in physics, typically specified with graph coordinates Introduction Position

Transcription

1 .0 - Introduction Object move: Ball bounce, car peed, and pacehip accelerate. We are o familiar with the concept of motion that we ue ophiticated phyic term in everyday language. For example, we might ay that a project ha reached ecape velocity or, if it i going le well, that it i in free fall. In thi chapter, you will learn more about motion, a field of tudy called kinematic. You will become familiar with concept uch a velocity, acceleration and diplacement. For now, the focu i on how thing move, not what caue them to move. Later, you will tudy dynamic, which center on force and how they affect motion. Dynamic and kinematic make up mechanic, the tudy of force and motion. Two key concept in thi chapter are velocity and acceleration. Velocity i how fat omething i moving (it peed) and in what direction it i moving. Acceleration i the rate of change in velocity. In thi chapter, you will have many opportunitie to learn about velocity and acceleration and how they relate. To get a feel for thee concept, you can experiment by uing the two imulation on the right. Thee imulation are verion of the tortoie and hare race. In thi claic parable, the teady tortoie alway win the race. With your help, though, the hare tand a chance. (After all, thi i your phyic coure, not your literature coure.) In the firt imulation, the tortoie ha a head tart and move at a contant velocity of three meter per econd to the right. The hare i initially tationary; it ha zero velocity. You et it acceleration in other word, how much it velocity change each econd. The acceleration you et i contant throughout the race. Can you et the acceleration o that the hare croe the finih line firt and win the race? To try, click on Interactive 1, enter an acceleration value in the entry box in the imulation, and pre GO to ee what happen. Pre RESET if you want to try again. Try acceleration value up to 10 meter per econd quared. (At thi acceleration, the velocity increae by 10 meter per econd every econd. Value larger than thi will caue the action to occur o rapidly that the hare may quickly diappear off the creen.) It doe not really matter if you can caue the hare to beat thi rather fat-moving tortoie. However, we do want you to try a few different rate of acceleration and ee how they affect the hare velocity. Nothing particularly tricky i occurring here; you are imply oberving two baic propertie of motion: velocity and acceleration. In the econd imulation, the race i a round trip. To win the race, a contetant need to go around the pot on the right and then return to the tarting line. The tortoie ha been given a head tart in thi race. When you tart the imulation, the tortoie ha already rounded the pot and i moving at a contant velocity on the hometretch back to the finih line. In thi imulation, when you pre GO the hare tart off moving quickly to the right. Again, you upply a value for it acceleration. The challenge i to upply a value for the hare' acceleration o that it turn around at the pot and race back to beat the tortoie. (Hint: Think negative! Acceleration can be either poitive or negative.) Again, it doe not matter if you win; we want you to notice how acceleration affect velocity. Doe the hare' velocity ever become zero? Negative? To anwer thee quetion, click on Interactive, enter the acceleration value for the hare in the gauge, pre GO to ee what happen, and RESET to try again. You can alo ue PAUSE to top the action and ee the velocity at any intant. Pre PAUSE again to retart the race. We have given you a fair number of concept in thi introduction. Thee fundamental are the foundation of the tudy of motion, and you will learn much more about them hortly..1 - Poition Poition: The location of an object; in phyic, typically pecified with graph coordinate. Poition tell you location. There are many way to decribe location: Beverly Hill 9010;...a galaxy far, far away ; a far away from you a poible. Each work in it own context. Phyicit often ue number and graph intead of word and phrae. Number and graph enable them (and you) to analyze motion with preciion and conitency. In thi chapter, we will analyze object that move in one dimenion along a line, like a train moving along a flat, Copyright Kinetic Book Co. Chapter 31

2 traight ection of track. To begin, we meaure poition along a number line. Two toy figure and a number line are hown in the illutration to the right. A you can ee, the zero point i called the origin. Poitive number are on the right and negative number are on the left. By convention, we draw number line from left to right. The number line could reflect an object' poition in eat and wet direction, or north and outh, or up and down; the important idea i that we can pecify poition by referring to point on a line. When an object move in one dimenion, you can pecify it poition by it location on the number line. The variable x pecifie that poition. For example, a hown in the illutration for Equation 1, the hiker tand at poition x = 3.0 meter and the toddler i at poition x =.0 meter. Later, you will tudy object that move in multiple dimenion. For example, a baketball free throw will initially travel both up and forward. For now, though, we will conider object that move in one dimenion. Poition Location of an object Relative to origin Poition x repreent poition Unit: meter (m) What are the poition of the figure? Hiker: x =.0 m Toddler: x = 3.0 m. - Diplacement Diplacement: The direction and ditance of the hortet path between an initial and final poition. You ue the concept of ditance every day. For example, you are told a home run travel 400 ft (1 meter) or you run the metric mile (1.5 km) in track (or happily watch other run a metric mile). Diplacement add the concept of direction to ditance. For example, you go approximately 954 mi (1540 km) outh when you travel from Seattle to Lo Angele; the ummit of Mount Everet i 9,035 ft ( meter) above ea level. (You may have noticed we are uing both metric and Englih unit. We will do thi only for the firt part of thi chapter, with the thought that thi may prove helpful if you are familiarizing yourelf with the metric ytem.) Diplacement Ditance and direction Meaure net change in poition Sometime jut ditance matter. If you want to be a million mile away from your 3 Copyright Kinetic Book Co. Chapter

3 younger brother, it doe not matter whether that eat, north, wet or outh. The ditance i called the magnitude the amount of the diplacement. Direction, however, can matter. If you walk 10 block north of your home, you are at a different location than if you walk 10 block outh. In phyic, direction often matter. For example, to get a ball to the ground from the top of a tall building, you can imply drop the ball. Throwing the ball back up require a very trong arm. Both the direction and ditance of the ball movement matter. The definition of diplacement i precie: the direction and length of the hortet path from the initial to the final poition of an object motion. A you may recall from your mathematic coure, the hortet path between two point i a traight line. Phyicit ue arrow to indicate the direction of diplacement. In the illutration to the right, the arrow point in the direction of the moue diplacement. Phyicit ue the Greek letter (delta) to indicate a change or difference. A change in poition i diplacement, and ince x repreent poition, we write x to indicate diplacement. You ee thi notation, and the equation for calculating diplacement, to the right. In the equation, x f repreent the final poition (the ubcript f tand for final) and x i repreent the initial poition (the ubcript i tand for initial). Diplacement i a vector. A vector i a quantity that mut be tated in term of it direction and it magnitude. Magnitude mean the ize or amount. Move five meter to the right i a decription of a vector. Scalar, on the other hand, are quantitie that are tated olely in term of magnitude, like a dozen egg. There i no direction for a quantity of egg, jut an amount. Diplacement x = x f x i x = diplacement x f = final poition x i = initial poition Unit: meter (m) In one dimenion, a poitive or negative ign i enough to pecify a direction. A mentioned, number to the right of the origin are poitive, and thoe to the left are negative. Thi mean diplacement to the right i poitive, and to the left it i negative. For intance, you can ee in Example 1 that the moue car tart at the poition +3.0 meter and move to the left to the poition 1.0 meter. (We meaure the poition at the middle of the car.) Since it move to the left 4.0 meter, it diplacement i 4.0 meter. Diplacement meaure the ditance olely between the beginning and end of motion. We can ue dance to illutrate thi point. Let ay you are dancing and you take three tep forward and two tep back. Although you moved a total of five tep, your diplacement after thi maneuver i one tep forward. It would be better to ue ign to decribe the dance direction, o we could decribe forward a poitive and backward a negative. Three tep forward and two tep back yield a diplacement of poitive one tep. Since diplacement i in part a meaure of ditance, it i meaured with unit of length. Meter are the SI unit for diplacement. What i the moue car diplacement? x = x f x i x = 1.0 m 3.0 m x = 4.0 m.3 - Velocity Velocity: Speed and direction. You are familiar with the concept of peed. It tell you how fat omething i going: 55 mile per hour (mi/h) i an example of peed. The peedometer in a car meaure peed but doe not indicate direction. When you need to know both peed and direction, you ue velocity. Velocity i a vector. It i the meaure of how fat and in which direction the motion i occurring. It i repreented by v. In thi ection, we focu on average velocity, which i repreented by v with a bar over it, a hown in Equation 1. A police officer ue the concept of both peed and velocity in her work. She might iue a ticket to a motorit for driving 36 mi/h (58 km/h) in a chool zone; in thi cae, peed matter but direction i irrelevant. In another ituation, he might be told that a upect i fleeing north on I-405 at 90 mi/h (149 km/h); now velocity i important becaue it tell her both how fat and in what direction. Velocity Speed and direction To calculate an object average velocity, divide it diplacement by the time it take to move that diplacement. Thi time i called the elaped time, and i repreented by t. The direction for velocity i the ame a for the diplacement. For intance, let ay a car move poitive 50 mi (80 km) between the hour of 1 P.M. and 3 P.M. It diplacement i poitive 50 mi, and two hour elape a it move that ditance. The car average velocity equal +50 mile divided by two hour, or +5 mi/h (+40 km/h). Note that the direction i poitive becaue the diplacement wa poitive. If the diplacement were negative, then the velocity would alo be negative. At thi point in the dicuion, we are intentionally ignoring any variation in the car velocity. Perhap the car move at contant peed, or Copyright Kinetic Book Co. Chapter 33

4 maybe it move fater at certain time and then lower at other. All we can conclude from the information above i that the car average velocity i +5 mi/h. Velocity ha the dimenion of length divided by time; the unit are meter per econd (). Velocity x = diplacement t = elaped time Unit: meter/econd () What i the moue velocity?.4 - Average velocity Average velocity: Diplacement divided by elaped time. Average velocity equal diplacement divided by the time it take for the diplacement to occur. For example, if it take you two hour to move poitive 100 mile (160 kilometer), your average velocity i +50 mi/h (80 km/h). Perhap you drive a car at a contant velocity. Perhap you drive really fat, low down for ruh-hour traffic, drive fat again, get pulled over for a ticket, and then drive at a moderate peed. In either cae, becaue your diplacement i 100 mi and the elaped time i two hour, your average velocity i +50 mi/h. Average velocity Diplacement divided by elaped time Since the average velocity of an object i calculated from it diplacement, you need to be able to tate it initial and final poition. In Example 1 on the right, you are hown the poition of three town and aked to calculate the average velocity of a trip. You mut calculate the diplacement from the initial to final poition to determine the average velocity. A claic phyic problem tempt you to err in calculating average velocity. The problem run like thi: A hiker walk one mile at two mile per hour, and the next mile at four mile per hour. What i the hiker' average velocity? If you average two and four and anwer that the average velocity i three mi/h, you will have erred. To anwer the problem, you mut firt calculate the elaped time. You cannot imply average the two velocitie. It take the hiker 1/ an hour to cover the firt mile, but only 1/4 an hour to walk the econd mile, for a total elaped time of 3/4 of an hour. The average velocity equal two mile divided by 3/4 of an hour, which i a little le than three mile per hour. 34 Copyright Kinetic Book Co. Chapter

5 Average velocity x = diplacement t = elaped time A plane flie Acme to Bend in.0 hr, then traight back to Cote in 1.0 hr. What i it average velocity for the trip in km/h?.5 - Intantaneou velocity Intantaneou velocity: Velocity at a pecific moment. Object can peed up or low down, or they can change direction. In other word, their velocity can change. For example, if you drop an egg off a 40-tory building, the egg velocity will change: It will move fater a it fall. Someone on the building 39 th floor would ee it pa by with a different velocity than would omeone on the 30 th. When we ue the word intantaneou, we decribe an object velocity at a particular intant. In Concept 1, you ee a naphot of a toy moue car at an intant when it ha a velocity of poitive ix meter per econd. The fable of the tortoie and the hare provide a claic example of intantaneou veru average velocity. A you may recall, the hare eemed fater becaue it could achieve a greater intantaneou velocity than could the tortoie. But the hare long nap meant that it average velocity wa le than that of the tortoie, o the tortoie won the race. When the average velocity of an object i meaured over a very hort elaped time, the reult i cloe to the intantaneou velocity. The horter the elaped time, the cloer the average and intantaneou velocitie. Imagine the egg falling pat the 39 th floor window in the example we mentioned earlier, and let ay you wanted to determine it intantaneou velocity at the midpoint of the window. You could ue a topwatch to time how long it take the egg to travel from the top to the bottom of the window. If you then divided the height of the window by the elaped time, Intantaneou velocity Velocity at a pecific moment Intantaneou velocity v = intantaneou velocity Copyright Kinetic Book Co. Chapter 35

6 the reult would be cloe to the intantaneou velocity. However, if you meaured the time for the egg to fall from 10 centimeter above the window midpoint to 10 centimeter below, and ued that diplacement and elaped time, the reult would be even cloer to the intantaneou velocity at the window midpoint. A you repeated thi proce "to the limit" meauring horter and horter ditance and elaped time x = diplacement t = elaped time (perhap uing motion enor to provide precie value) you would get value cloer and cloer to the intantaneou velocity. To decribe intantaneou velocity mathematically, we ue the terminology hown in Equation 1. The arrow and the word lim mean the limit a t approache zero. The limit i the value approached by the calculation a it i performed for maller and maller interval of time. To give you a ene of velocity and how it change, let again ue the example of the egg. We calculate the velocity at variou time uing an equation you may have not yet encountered, o we will jut tell you the reult. Let aume each floor of the building i four meter (13 ft) high and that the egg i being dropped in a vacuum, o we do not have to worry about air reitance lowing it down. One econd after being dropped, the egg will be traveling at 9.8 meter per econd; at three econd, it will be traveling at 9 ; at five econd, 49 (or 3 ft/, 96 ft/ and 160 ft/, repectively.) After even econd, the egg ha an intantaneou velocity of 0. Why? The egg hit the ground at about 5.7 econd and therefore i not moving. (We aume the egg doe not rebound, which i a reaonable aumption with an egg.) Phyicit uually mean intantaneou velocity when they ay velocity becaue intantaneou velocity i often more ueful than average velocity. Typically, thi i expreed in tatement like the velocity when the elaped time equal three econd..6 - Poition-time graph and velocity Poition-time graph Show poition of object over time Steeper graph = greater peed A graph of an object' poition over time i a ueful tool for analyzing motion. You ee uch a poition-time graph above. Value on the vertical axi repreent the moue car' poition, and time i plotted on the horizontal axi. You can ee from the graph that the moue car tarted at poition x = 4 m, then moved to the poition x = +4 m at about t = 4.5, tayed there for a couple of econd, and then reached the poition x = m again after a total of 1 econd of motion. Where the graph i horizontal, a at point B, it indicate the moue poition i not changing, which i to ay the moue i not moving. Where the graph i teep, poition i changing rapidly with repect to time and the moue i moving quickly. Diplacement and velocity are mathematically related, and a poition-time graph can be ued to find the average or intantaneou velocity of an object. The lope of a traight line between any two point of the graph i the object average velocity between them. Why i the average velocity the ame a thi lope? The lope of a line i calculated by dividing the change in the vertical direction by the change in the horizontal direction, the rie over the run. In a poition-time graph, the vertical value are the x poition and the horizontal value tell the time. The lope of the line i the change in poition, which i diplacement, divided by the change in time, which i the elaped time. Thi i the definition of average velocity: diplacement divided by elaped time. Average velocity Slope of line between two point You ee thi relationhip tated and illutrated in Equation 1. Since the lope of the line hown in thi illutration i poitive, the average velocity between the two point on the line i poitive. Since the moue move to the right between thee point, it diplacement i poitive, which confirm that it average velocity i poitive a well. The lope of the tangent line for any point on a traight-line egment of a poition-time graph i contant. When the lope i contant, the velocity i contant. An example of contant velocity i the horizontal ection of the graph that include the point B in the illutration above. The lope of a tangent line at different point on a curve i not contant. The lope at a ingle point on a curve i determined by the lope of the tangent line to the curve at that point. You ee a tangent line illutrated in Equation. The lope a meaured by the tangent line equal the intantaneou velocity at the point. The lope of the tangent Intantaneou velocity Slope of tangent line at point line in Equation i negative, o the velocity there i negative. At that point, the moue i moving from right to left. The negative diplacement over a hort time interval confirm that it velocity i negative. 36 Copyright Kinetic Book Co. Chapter

7 What i the average velocity between point A and B? Conider the point A, B and C. Where i the intantaneou velocity zero? Where i it poitive? Where i it negative? Zero at B Poitive at A Negative at C.7 - Interactive problem: draw a poition-time graph In thi ection, you are challenged to match a pre-drawn poition-time graph by moving a ball along a number line. A you drag the ball, it poition at each intant will be graphed. Your challenge i to get a cloe a you can to the target graph. When you open the interactive imulation on the right, you will ee a graph and a coordinate ytem with x poition on the vertical axi and time on the horizontal axi. Below the graph i a ball on a number line. Examine the graph and decide how you will move the ball over the 10 econd to bet match the target graph. You may find it helpful to think about the velocity decribed by the target graph. Where i it increaing? decreaing? zero? If you are not ure, review the ection on poitiontime graph and velocity. You can chooe to diplay a graph of the velocity of the motion of the ball a decribed by the target graph by clicking a checkbox. We encourage you to think firt about what the velocity will be and ue thi checkbox to confirm your hypothei. Create your graph by dragging the ball and watching the graph of it motion. You can pre RESET and try again a often a you like. Copyright Kinetic Book Co. Chapter 37

8 .8 - Interactive problem: match a graph uing velocity In the interactive imulation on the right, you ee a poition-time graph of a ball that move horizontally. The target graph ha three traight-line egment: from 0 to 3 econd, 3 to 7 econd, and 7 to 10 econd. You et the velocity for three time interval. Your challenge i to et the velocity for each interval to match the target graph. The velocity i contant for each of the interval. You can calculate each velocity from the lope of the appropriate line egment. Review the ection on poition-time graph and velocity if you are not ure how to do thi. Enter the velocity for each egment to the nearet meter per econd and pre GO to tart the ball moving. Pre RESET to tart over..9 - Velocity graph and diplacement If you graph an object' velocity veru time, the area between the graph and the horizontal axi equal the object' diplacement. The horizontal axi repreent time, t. The velocity graph hown in Concept 1 i a horizontal line that indicate an object moving at a contant velocity of To determine the object diplacement between.0 and 6.0 econd from the graph, we calculate the area of the rectangle hown in Concept 1, which equal 0 m. Why doe the area equal the diplacement? Becaue it equal the product of velocity and elaped time, which i diplacement. (Some algebra, uing the definition of velocity, yield thi reult: v = = x/ t, o x = v t.) In the econd graph, in Concept, we how the graph of an object whoe velocity change over time. Again, we can ue the principle that the area between the velocity graph and the horizontal time axi equal the diplacement. If we know the function decribing the velocity, we can calculate the area (and the diplacement) exactly uing calculu. Velocity-time graph Diplacement = area between graph and time axi Here, we approximate the area by uing a et of rectangle. The um of the area of the five vertical rectangle between the horizontal axi and the graph i a good approximation for the area under the graph. The area of each rectangle equal the product of a mall amount of elaped time (the width of the rectangle) and a velocity (the height of the graph at that point). By keeping the width of the rectangle mall, we enure that the height of each rectangle i cloe to the average velocity for that elaped time. Since the area i cloe to the average velocity time the elaped time, it i a good approximation of the diplacement for that period of time. The um of all the rectangular area then approximate the total diplacement. The um of the area of the rectangle in Concept i 0 meter. Uing calculu, we calculated the exact diplacement a 0.8 meter, o the etimate with rectangle i quite cloe. Remember that diplacement can be poitive or negative. When the velocity graph i above the horizontal axi, the velocity i poitive and the diplacement i poitive. When it i below, the velocity i negative and the diplacement during that interval i negative. You ee thi point emphaized in Concept 3. Velocity-time graph Area approximated by rectangle Velocity-time graph Negative diplacement below time axi 38 Copyright Kinetic Book Co. Chapter

9 What i the object diplacement in the firt ix econd? area below t axi = ½(.0 )( 1.0 ) area above t axi = ½(4.0 )(.0 ) x = 1.0 m m x = 3.0 m.10 - Acceleration Acceleration: Change in velocity. When an object velocity change, it accelerate. Acceleration meaure the rate at which an object peed up, low down or change direction. Any of thee variation contitute a change in velocity. The letter a repreent acceleration. Acceleration i a popular topic in port car commercial. In the commercial, acceleration i often expreed a how fat a car can go from zero to 60 mile per hour (97 km/h, or 7 ). For example, a current model Corvette automobile can reach 60 mi/h in 4.9 econd. There are even hotter car than thi in production. A racing car accelerate. To calculate average acceleration, divide the change in intantaneou velocity by the elaped time, a hown in Equation 1. To calculate the acceleration of the Corvette, divide it change in velocity, from 0 to 7, by the elaped time of 4.9 econd. The car accelerate at an average rate of 5.5 per econd. We typically expre thi a 5.5 meter per econd quared, or 5.5. (Thi equal 18 ft/, and with thi obervation we will ceae tating value in both meaurement ytem, in order to implify the expreion of number.) Acceleration i meaured in unit of length divided by time quared. Meter per econd quared ( ) expre acceleration in SI unit. Let aume the car accelerate at a contant rate; thi mean that each econd the Corvette move 5.5 fater. At one econd, it i moving at 5.5 ; at two econd, 11 ; at three econd, 16.5 ; and o forth. The car velocity increae by 5.5 every econd. Since acceleration meaure the change in velocity, an object can accelerate even while it i moving at a contant peed. For intance, conider a car moving around a curve. Even if the car peed remain contant, it accelerate becaue the change in the car direction mean it velocity (peed plu direction) i changing. Acceleration can be poitive or negative. If the Corvette ue it brake to go from +60 to 0 mi/h in 4.9 econd, it velocity i decreaing jut a fat a it wa increaing before. Thi i an example of negative acceleration. Acceleration Change in velocity You may want to think of negative acceleration a lowing down, but be careful! Let ay a train ha an initial velocity of negative 5 and that change to negative 50. The train i moving at a fater rate (peeding up) but it ha negative acceleration. To be precie, it negative acceleration caue an increaingly negative velocity. Acceleration Velocity and acceleration are related but ditinct value for an object. For example, an object can have poitive velocity and negative acceleration. In thi cae, it i lowing down. An object can have zero velocity, yet be accelerating. For example, when a ball bounce off the ground, it experience a moment of zero velocity a it velocity change Copyright Kinetic Book Co. Chapter 39

10 from negative to poitive, yet it i accelerating at thi moment ince it velocity i changing. v = change in intantaneou velocity t = elaped time Unit: meter per econd quared ( ) What i the average acceleration of the moue between.5 and 4.5 econd?.11 - Average acceleration Average acceleration: The change in intantaneou velocity divided by the elaped time. Average acceleration i the change in intantaneou velocity over a period of elaped time. It definition i hown in Equation 1 to the right. We will illutrate average acceleration with an example. Let' ay you are initially driving a car at 1 meter per econd and 8 econd later you are moving at 16. The change in velocity i 4 during that time; the elaped time i eight econd. Dividing the change in velocity by the elaped time determine that the car accelerate at an average rate of 0.5. Perhap the car' acceleration wa greater during the firt four econd and le during the lat four econd, or perhap it wa contant the entire eight econd. Whatever the cae, the average acceleration i the ame, ince it i defined uing the initial and final intantaneou velocitie. Average acceleration Change in intantaneou velocity divided by elaped time Average acceleration v = change in intantaneou velocity t = elaped time 40 Copyright Kinetic Book Co. Chapter

11 What i the moue average acceleration?.1 - Intantaneou acceleration Intantaneou acceleration: Acceleration at a particular moment. You have learned that velocity can be either average or intantaneou. Similarly, you can determine the average acceleration or the intantaneou acceleration of an object. We ue the moue in Concept 1 on the right to how the ditinction between the two. The moue move toward the trap and then wiely turn around to retreat in a hurry. The illutration how the moue a it move toward and then hurrie away from the trap. It tart from a ret poition and move to the right with increaingly poitive velocity, which mean it ha a poitive acceleration for an interval of time. Then it low to a top when it ee the trap, and it poitive velocity decreae to zero (thi i negative acceleration). It then move back to the left with increaingly negative velocity (negative acceleration again). If you would like to ee thi action occur again in the Concept 1 graphic, pre the refreh button in your brower. We could calculate an average acceleration, but decribing the moue' motion with intantaneou acceleration i a more informative decription of that motion. At ome intant in time, it ha poitive acceleration and at other intant, negative acceleration. By knowing it acceleration and it velocity at an intant in time, we can determine whether it i moving toward the trap with increaingly poitive velocity, lowing it rate of approach, or moving away with increaingly negative velocity. Intantaneou acceleration i defined a the change in velocity divided by the elaped time a the elaped time approache zero. Thi concept i tated mathematically in Equation 1 on the right. Earlier, we dicued how the lope of the tangent at any point on a poition-time graph equal the intantaneou velocity at that point. We can apply imilar reaoning here to conclude that the intantaneou acceleration at any point on a velocity-time graph equal the lope of the tangent, a hown in Equation. Why? Becaue lope equal the rate of change, and acceleration i the rate of change of velocity. Intantaneou acceleration Acceleration at a particular moment Intantaneou acceleration a = intantaneou acceleration v = change in velocity t = elaped time (approache 0) In Example 1, we how a graph of the velocity of the moue a it approache the trap and then flee. You are aked to determine the ign of the intantaneou acceleration at four point; you can do o by conidering the lope of the tangent to the velocity graph at each point. Intantaneou acceleration Slope of line tangent to point on velocity-time graph Copyright Kinetic Book Co. Chapter 41

12 The graph how the moue' velocity veru time. Decribe the intantaneou acceleration at A, B, C and D a poitive, negative or zero. a poitive at A a negative at B a zero at C a negative at D.13 - Interactive problem: tortoie and hare candal The official at a race have learned that one of the contetant ha been eating illegal performance-enhancing vegetable. If either animal exceed it previou record, it ha cheated. Your job i to identify the cheater. It i well known among track enthuiat that the tortoie alway run at a contant velocity and the hare alway run with a contant acceleration. The tortoie peronal bet i The hare mot rapid acceleration ha been If either animal beat it previou bet, it i the guilty party in the vegetable candal. The hare give the tortoie a head tart of 31.0 meter to make the race more intereting. Click on the graphic on the right and pre GO to tart the race. You will ee value for the poition of the tortoie and the velocity of the hare in the control panel. You will need to record thee value at a couple time in the race. Pre PAUSE during the race to do o. Pre PAUSE to reume the race a well. From the value you record, calculate the velocity of the tortoie and the acceleration of the hare. Who i cheating? If you decide the tortoie cheated, click on the lettuce; if the culprit i the hare, click on the carrot. The imulation will confirm (or reject) your concluion. Pre RESET and GO to run the race again if you need to. If you have trouble anwering the quetion, refer to the ection on velocity and acceleration, and look at the equation that define thee term Sample problem: velocity and acceleration The moue car goe 10.3 meter in 4.15 econd at a contant velocity, then accelerate at 1. for 5.34 more econd. What i it final velocity? Solving thi problem require two calculation. The moue car' velocity during the firt part of it journey mut be calculated. Uing that value a the initial velocity of the econd part of the journey, and the rate of acceleration during that part, you can calculate the final velocity. 4 Copyright Kinetic Book Co. Chapter

13 Draw a diagram Variable Part 1: Contant velocity diplacement elaped time velocity x = 10.3 m t = 4.15 v Part : Contant acceleration initial velocity v i = v (calculated above) acceleration a = 1. elaped time final velocity t = 5.34 v f What i the trategy? 1. Ue the definition of velocity to find the velocity of the moue car before it accelerate. The velocity i contant during the firt part of the journey.. Ue the definition of acceleration and olve for the final velocity. Phyic principle and equation The definition of velocity and acceleration will prove ueful. The velocity and acceleration are contant in thi problem. In thi and later problem, we ue the definition for average velocity and acceleration without the bar over the variable. v = x/ t a = v/ t = (v f v i )/ t Step-by-tep olution We tart by finding the velocity before the engine fire. Step Reaon 1. v = x/ t definition of velocity. v = (10.3 m)/(4.15 ) enter value 3. v =.48 divide Next we find the final velocity uing the definition of acceleration. The initial velocity i the ame a the velocity we jut calculated. Step Reaon 4. a = (v f v i )/ t definition of acceleration 5. enter given value, and velocity from tep = v f.48 multiply by v f = 8.99 olve for v f Copyright Kinetic Book Co. Chapter 43

14 .15 - Interactive checkpoint: ubway train A ubway train accelerate along a traight track at a contant How long doe it take the train to increae it peed from 4.47 to 13.4? Anwer: t =.16 - Interactive problem: what wrong with the rabbit? You jut bought five rabbit. They were uppoed to be contant acceleration rabbit, but you worry that ome are the le expenive, non-contant acceleration rabbit. In fact, you think two might be the cheaper critter. You take them home. When you pre GO, they will run or jump for five econd (well, one jut it till) and then the imulation top. You can pre GO a many time a you like and ue the PAUSE button a well. Your miion: Determine if you were ripped off, and drag the ½ off ale tag to the cheaper rabbit. The imulation will let you know if you are correct. You may decide to keep the cuddly creature, but you want to be fairly charged. Each rabbit ha a velocity gauge that you can ue to monitor it motion in the imulation. The implet way to olve thi problem i to conider the rabbit one at a time: look at a rabbit velocity gauge and determine if the velocity i changing at a contant rate. No detailed mathematical calculation are required to olve thi problem. If you find thi imulation challenging, focu on the relationhip between acceleration and velocity. With a contant rate of acceleration, the velocity mut change at a contant rate: no jump or udden change. Hint: No change in velocity i zero acceleration, a contant rate Derivation: creating new equation Other ection in thi chapter introduced ome of the fundamental equation of motion. Thee equation defined fundamental concept; for example, average velocity equal the change in poition divided by elaped time. Several other helpful equation can be derived from thee baic equation. Thee equation enable you to predict an object motion without knowing all the detail. In thi ection, we derive the formula hown in Equation 1, which i ued to calculate an object final velocity when it initial velocity, acceleration and diplacement are known, but not the elaped time. If the elaped time were known, then the final velocity could be calculated uing the definition of velocity, but it i not. Thi equation i valid when the acceleration i contant, an aumption that i ued in many problem you will be poed. Variable Deriving a motion equation v f + a x v i = initial velocity v f = final velocity a = contant acceleration x = diplacement We ue t intead of t to indicate the elaped time. Thi i impler notation, and we will ue it often. acceleration (contant) a initial velocity v i final velocity elaped time diplacement v f t x 44 Copyright Kinetic Book Co. Chapter

15 Strategy Firt, we will dicu our trategy for thi derivation. That i, we will decribe our overall plan of attack. Thee trategy point outline the major tep of the derivation. 1. We tart with the definition of acceleration and rearrange it. It include the term for initial and final velocity, a well a elaped time.. We derive another equation involving time that can be ued to eliminate the time variable from the acceleration equation. The condition of contant acceleration will be crucial here. 3. We eliminate the time variable from the acceleration equation and implify. Thi reult in an equation that depend on other variable, but not time. Phyic principle and equation Since the acceleration i contant, the velocity increae at a contant rate. Thi mean the average velocity i the um of the initial and final velocitie divided by two. We will ue the definition of acceleration, a = (v f v i )/t We will alo ue the definition of average velocity, Step-by-tep derivation We tart the derivation with the definition of average acceleration, olve it for the final velocity and do ome algebra. Thi create an equation with the quare of the final velocity on the left ide, where it appear in the equation we want to derive. Step Reaon 1. a = (v f v i )/t definition of average acceleration. v f + at olve for final velocity 3. v f = (v i + at) quare both ide 4. v f + v i at + a t expand right ide 5. v f + at(v i + at) factor out at 6. v f + at(v i + v i + at) rewrite v i a a um 7. v f + at(v i + v f ) ubtitution from equation The equation we jut found i the baic equation from which we will derive the deired motion equation. But it till involve the time variable t multiplied by a um of velocitie. In the next tage of the derivation, we ue two different way of expreing the average velocity to develop a econd equation involving time multiplied by velocitie. We will ubequently ue that econd equation to eliminate time from the equation above. Step Reaon 8. average velocity i average of initial and final velocitie 9. definition of average velocity 10. et right ide of 8 and 9 equal 11. t(v i + v f ) = x rearrange equation Copyright Kinetic Book Co. Chapter 45

16 We have now developed two equation that involve time multiplied by a um of velocitie. The left ide of the equation in tep 11 matche an expreion appearing in equation 7, at the end of the firt tage. By ubtituting from thi equation into equation 7, we eliminate the time variable t and derive the deired equation. Step Reaon 1. v f + a( x) ubtitute right ide of 11 into v f + a x rearrange factor We have now accomplihed our goal. We can calculate the final velocity of an object when we know it initial velocity, it acceleration and it diplacement, but do not know the elaped time. The derivation i finihed Motion equation for contant acceleration The equation above can be derived from the fundamental definition of motion (equation uch a a = v/ t). To undertand the equation, you need to remember the notation: x for diplacement, v for velocity and a for acceleration. The ubcript i and f repreent initial and final value. We follow a common convention here by uing t for elaped time intead of t. We how the equation above and below on the right. Note that to hold true thee equation all require a contant rate of acceleration. Analyzing motion with a varying rate of acceleration i a more challenging x = diplacement, v = velocity, a = acceleration, t = elaped time tak. When we refer to acceleration in problem, we mean a contant rate of acceleration unle we explicitly tate otherwie. To olve problem uing motion equation like thee, you look for an equation that include the value you know, and the one you are olving for. Thi mean you can olve for the unknown variable. In the example problem to the right, you are aked to determine the acceleration required to top a car that i moving at 1 meter per econd in a ditance of 36 meter. In thi problem, you know the initial velocity, the final velocity (topped = 0.0 ) and the diplacement. You do not know the elaped time. The third motion equation include the two velocitie, the acceleration, and the diplacement, but doe not include the time. Since thi equation include only one value you do not know, it i the appropriate equation to chooe. Applying motion equation Determine the known and the unknown() Find other known from ituation Chooe an equation with thoe variable Motion equation 46 Copyright Kinetic Book Co. Chapter

17 What acceleration will top the car exactly at the top ign? v f + a x a = (v f v i )/ x a = 144/7 a = Sample problem: a printer What i the runner' velocity at the end of a 100-meter dah? You are aked to calculate the final velocity of a printer running a 100-meter dah. Lit the variable that you know and the one you are aked for, and then conider which equation you might ue to olve the problem. You want an equation with jut one unknown variable, which in thi problem i the final velocity. The printer initial velocity i not explicitly tated, but he tart motionle, o it i zero. Draw a diagram Variable diplacement x = 100 m acceleration a = 0.58 initial velocity final velocity v i = 0.00 v f What i the trategy? 1. Chooe an appropriate equation baed on the value you know and the one you want to find.. Enter the known value and olve for the final velocity. Phyic principle and equation Baed on the known and unknown value, the equation below i appropriate. We know all the variable in the equation except the one we are aked to find, o we can olve for it. v f + a x Copyright Kinetic Book Co. Chapter 47

18 Step-by-tep olution Step Reaon 1. v f = v i + a x motion equation. enter known value 3. v f = 106 m / multiplication and addition 4. v f = 10.3 take quare root In tep 4, we take the quare root of 106 to find the final velocity. We choe the poitive quare root, ince the runner i moving in the poitive direction. When there are multiple root, you look at the problem to determine the olution that make ene given the circumtance. If the runner were running to the left, then a negative velocity would be the appropriate choice..0 - Sample problem: initial and final velocity The moue goe 11.8 meter in 3.14 econd at a contant acceleration of 1.1. What i it velocity at the beginning and end of the 11.8 meter? To olve thi problem, lit the known variable and the one you are aked for. Since there are two unknown value, the initial and final velocity, you will need to ue two equation to olve thi problem. Variable diplacement x = 11.8 m acceleration a = 1.1 elaped time initial velocity final velocity t = 3.14 v i v f What i the trategy? 1. There are two unknown, the initial and final velocitie, o chooe two equation that include thee two unknown and the value you do know.. Subtitute known value and ue algebra to reduce the two equation to one equation with a ingle unknown value. 3. Solve the reduced equation for one of the unknown value, and then calculate the other value. Phyic principle and equation Thee two motion equation contain the known and unknown value, and no other value. v f + at x = ½(v i + v f )t Step-by-tep olution We tart with a motion equation containing the two velocitie we want to find, and ubtitute known value, and implify. Step Reaon 1. v f + at firt motion equation. v f + (1.1 )(3.14 ) ubtitute value 3. v f multiply 48 Copyright Kinetic Book Co. Chapter

19 Now we ue a econd motion equation containing the two velocitie, ubtitute known value, and implify. Thi give u two equation with the two unknown we want to find. Step Reaon 4. x = ½(v i + v f )t econd motion equation m = ½(v i + v f )(3.14 ) ubtitute known value v f multiply by, divide by 3.14 Now we olve the two equation. Step Reaon v i ubtitute equation 3 into equation 6 8. v i = 1.86 olve for v i 9. v f = 5.66 from equation 3 There are other way to olve thi problem. For example, you could ue the equation x t + ½at to find the initial velocity from the diplacement, acceleration, and elaped time. Then you could ue the equation v f + at to olve for the final velocity..1 - Interactive checkpoint: paenger jet A paenger jet land on a runway with a velocity of Once it touche down, it accelerate at a contant rate of How far doe the plane travel down the runway before it velocity i decreaed to.00, it taxi peed to the landing gate? Anwer: x = m. - Interactive problem: tortoie and hare meet again The tortoie and the hare are at it again. Thi time, the race i a round trip. The runner have to go out, turn around a pot, and return to the tarting point. Your miion i to make the hare win. The tortoie ha a head tart of 40.0 meter and i ticking to it trategy of moving at a contant peed. The hare' trategy i to tart with a large poitive velocity and accelerate o it turn around after paing the pot. The hare' initial velocity i The pot i 50.0 meter from the tarting line, and the hare need one meter for it turn, o it need to turn around at 51.0 m. To put it another way: It need to have zero velocity at thi point. Click on the graphic on the right and et the hare' acceleration to the nearet 0.01, then pre GO to ee the race. Pre RESET to try again. If you have difficulty getting the hare to win, refer back to the motion equation ection. Copyright Kinetic Book Co. Chapter 49

20 .3 - Free-fall acceleration Free-fall acceleration: Rate of acceleration due to the force of Earth' gravity. Galileo Galilei i reputed to have conducted an intereting experiment everal hundred year ago. According to legend, he dropped two ball with different mae off the Leaning Tower of Pia and found that both landed at the ame time. Their differing mae did not change the time it took them to fall. (We ay he wa reputed to have becaue there i little evidence that he in fact conducted thi experiment. He wa more of a roll ball down a plane experimenter.) Today thi experiment i ued to demontrate that free-fall acceleration i contant: that the acceleration of a falling object due olely to the force of gravity i contant, regardle of the object ma or denity. The two ball landed at the ame time becaue they tarted with the ame initial velocity, traveled the ame ditance and accelerated at the ame rate. In 1971, the commander of Apollo 15 conducted a verion of the experiment on the Moon, and demontrated that in the abence of air reitance, a hammer and a feather accelerated at the ame rate and reached the urface at the ame moment. Free-fall acceleration Acceleration due to gravity In Concept 1, you ee a photograph that illutrate free-fall acceleration. Picture of a freely falling egg were taken every /15 of a econd. Since the egg peed contantly increae, the ditance between the image increae over time. Greater diplacement over the ame interval of time mean it velocity i increaing in magnitude; it i accelerating. Free-fall acceleration i the acceleration caued by the force of the Earth gravity, ignoring other factor like air reitance. It i ometime tated a the rate of acceleration in a vacuum, where there i no air reitance. Near the Earth urface, it magnitude i 9.80 meter per econd quared. The letter g repreent thi value. The value of g varie lightly baed on location. It i le at the Earth' pole than at the equator, and i alo le atop a tall mountain than at ea level. Galileo' famou experiment Confirmed by Apollo 15 on the Moon The acceleration of 9.80 occur in a vacuum. In the Earth atmophere, a feather and a mall lead ball dropped from the ame height will not land at the ame time becaue the feather, with it greater urface area, experience more air reitance. Since it ha le ma than the ball, gravity exert le force on it to overcome the larger air reitance. The acceleration will alo be different with two object of the ame ma but different urface area: A flat heet of paper will take longer to reach the ground than the ame heet crumpled up into a ball. By convention, up i poitive, and down i negative, like the value on the y axi of a graph. Thi mean when uing g in problem, we tate free-fall acceleration a negative To make thi ditinction, we typically ue a or a y when we are uing the negative ign to indicate the direction of free-fall acceleration. Free-fall acceleration occur regardle of the direction in which an object i moving. For example, if you throw a ball traight up in the air, it will low down, accelerating at 9.80 until it reache zero velocity. At that point, it will then begin to fall back toward the ground and continue to accelerate toward the ground at the ame rate. Thi mean it velocity will become increaingly negative a it move back toward the ground. Free-fall acceleration on Earth g = 9.80 g = magnitude of free-fall acceleration The two example problem in thi ection tre thee point. For intance, Example on the right ak you to calculate how long it will take a ball thrown up into the air to reach it zero velocity point (the peak of it motion) and it acceleration at that point. What i the egg' velocity after falling from ret for 0.10 econd? v f + at 50 Copyright Kinetic Book Co. Chapter