LINEAR MOTION
Objectives • Explain how you can tell an object is moving. (4.1) 4 LINEAR MOTION • Describe how you can calculate speed. (4.2) You can describe the motion of • Distinguish between speed and THE BIG velocity. (4.3) an object by its position, speed,
• Describe how you can calculate IDEA ...... direction, and acceleration. acceleration. (4.4) • Describe the acceleration of an object in free fall. (4.5) ore than 2000 years ago, the ancient • Describe how the distance Greek scientists were familiar with fallen per second changes for some of the ideas of physics that an object in free fall. (4.6) M we study today. They had a good under- • Describe what the slope of standing of some of the properties of a speed-versus-time graph represents. (4.7) light, but they were confused about motion. Great progress in understand- • Describe how air resistance affects the motion of falling ing motion occurred with Galileo and objects. (4.8) his study of balls rolling on inclined • Explain the relationship planes, as discussed in the previous between velocity and chapter. In this chapter, we look at acceleration. (4.9) motion in more detail.
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MATERIALS washers, string, pie tin, meter stick EXPECTED OUTCOME The time between the sounds will be discover! the same. Do Objects Fall Faster the Longer Analyze and Conclude ANALYZE AND CONCLUDE They Fall? 1. Observing Describe the time intervals 1. The time between the 1. Attach washers to a 3.5 m long string at the between sounds. sounds was the same. following distances from one end: 0.0 m, 2. Predicting What would you hear if the wash- 2. The sounds would be heard 0.11 m, 0.44 m, 0.99 m, 1.76 m and 2.76 m. ers were evenly spaced on the string? at progressively shorter time intervals. 2. Stand on a chair or desk with the string and 3. Making Generalizations What can you say washers suspended over a piece of metal 3. The distance traveled about the distance traveled by a falling object such as a pie tin. The washer tied to the end during each second of fall during each second of fall? of the string should be just touching the increases. metal surface. TEACHING TIP The distance 3. Release the string and listen to the rate at between masses was calculated using d 5 1/2gt 2 which the washers hit the metal. You may with time intervals of 0.15 s. wish to perform a second trial to confirm your observations.
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4.1 Motion Is Relative Everything moves. Even things that appear to be at rest move. They think! move with respect to the sun and stars. When we describe the motion A hungry mosquito sees 4.1 Motion is of one object with respect to another, we say that the object is moving you resting in a hammock Relative in a 3-meter per second relative to the other object. A book that is at rest, relative to the table Key Term it lies on, is moving at about 30 kilometers per second relative to the breeze. How fast and in what direction should relative sun. The book moves even faster relative to the center of our galaxy. the mosquito fly in order Teaching Tip Begin with When we discuss the motion of something, we describe its motion to hover above you for a description of motion that relative to something else. An object is moving if its position lunch? will be more quantitative than relative to a fixed point is changing. When we say that a space Answer: 4.1 later material. This idea that shuttle moves at 8 kilometers per second, we mean its movement the course does not become progressively more difficult relative to Earth below. When we say a racing car in the Indy 500 should lessen anxiety for students reaches a speed of 300 kilometers per hour, of course we mean rela- who have read Chapter 4 and tive to the track. Unless stated otherwise, when we discuss the speeds found it intimidating. Spend less of things in our environment, we mean speed with respect to the time on Chapter 4 to have time surface of Earth even though Earth moves around the sun, as shown for more interesting physics later! in Figure 4.1. Motion is relative. CONCEPT How can you tell if an object is moving?
CHECK ......
CONCEPT An object is moving
CHECK ...... if its position relative FIGURE 4.2 to a fixed point is changing. The racing cars in the Indy 500 move relative to the track. Teaching Resources • Reading and Study Workbook • PresentationEXPRESS • Interactive Textbook • Conceptual Physics Alive! DVDs Linear Motion
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Key Terms 4.2 Speed speed, instantaneous speed, think! Before the time of Galileo, people described moving things as simply average speed If a cheetah can maintain a “slow” or “fast.” Such descriptions were vague. Galileo is credited as Teaching Tip Define speed constant speed of 25 m/s, being the first to measure speed by considering the distance covered by writing speed 5 distance/time it will cover 25 meters and the time it takes. Speed is how fast an object is moving. You on the board while giving every second. At this rate, can calculate the speed of an object by dividing the distance examples—automobile how far will it travel in speedometers, etc. Use the delta 10 seconds? In 1 minute? covered by time. symbol (D) from the beginning, distance Answer: 4.2.1 Speed ϭ even though the text waits until time later. So speed 5 Ddistance/Dtime. Stress that the slash in your For example, if a cheetah, such as the one shown in Figure 4.3, notation is a division sign that means “per”—as in so many covers 50 meters in a time of 2 seconds, its speed is 25 m/s. kilometers per so many hours. Then walk across the floor with FIGURE 4.3 1-m strides once per second and A cheetah is the fastest explain that you cover a distance land animal over dis- of 1 meter per second; hence tances less than 500 your speed is 1 m/s. meters and can achieve peak speeds of 100 km/h. Ask If you ride a bike a distance of 5 m in 1 s, what is your speed? What is your speed for 10 m in 2 s? For 100 m in 20 s? Each answer is 5 m/s. Teaching Tip You’ll appreciably improve your instruction if you allow some thinking time (say 3 s) after you ask a question. If no one answers Any combination of units for distance and time that are useful in 5 s, your class may need an easier question first. and convenient are legitimate for describing speed. Miles per hour (mi/h), kilometers per hour (km/h), centimeters per day (the speed Teaching Tidbit Faster than the cheetah is the cosmopolitan of a sick snail?), or light-years per century are all legitimate units for sailfish, the fastest fish over short speed. The slash symbol (/) is read as “per.” Throughout this book, distances (>100 km/h). we’ll primarily use the units meters per second (m/s) for speed. Table 4.1 shows some comparative speeds in different units.
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0046_CP09_SE_CH04.indd 48 11/12/07 5:39:11 PM 0046_CP09_SE_CH04.indd 49 00046_cp09te_CH04.indd046_cp09te_CH04.indd 4488 11/30/08/30/08 111:28:331:28:33 AMAM High school students easily Instantaneous Speed A car does not always move at the confuse the concepts of speed, same speed. A car may travel down a street at 50 km/h, slow velocity, and acceleration to 0 km/h at a red light, and speed up to only 30 km/h because the concepts are difficult to distinguish and because of traffic. You can tell the speed of the car at any because they usually appear instant by looking at the car’s speedometer, such as the early in the course. Beware of one in Figure 4.4. The speed at any instant is called spending too much time on the instantaneous speed. A car traveling at 50 km/h Chapter 4. Make the distinctions among speed, velocity, and may go at that speed for only one minute. If the car acceleration and move on. Don’t continued at that speed for a full hour, it would cover get bogged down in Chapter 4. 50 km. If it continued at that speed for only half an hour, it would cover only half that distance, or 25 km. In one minute, the car would cover less than 1 km. FIGURE 4.4 Average Speed In planning a trip by car, the driver often wants The speedometer for a to know how long it will take to cover a certain distance. The car North American car gives will certainly not travel at the same speed all during the trip. The readings of instantaneous driver cares only about the average speed for the trip as a whole. speed in both mi/h and The average speed is the total distance covered divided by the time. km/h. Odometers for the U.S. market give readings total distance covered average speed ϭ in miles; those for the time interval Canadian market give Average speed can be calculated rather easily. For example, if we readings in kilometers. drive a distance of 60 kilometers during a time of 1 hour, we say our average speed is 60 kilometers per hour (60 km/h). Or, if we travel 240 kilometers in 4 hours, total distance covered 240 km average speed ϭ ϭ ϭ 60 km/h time interval 4 h Note that when a distance in kilometers (km) is divided by a time in hours (h), the answer is in kilometers per hour (km/h). Since average speed is the distance covered divided by the time of travel, it does not indicate variations in the speed that may take place during the trip. In practice, we experience a variety of speeds on most trips, so the average speed is often quite different from the instanta- CONCEPT You can calculate
neous speed. Whether we talk about average speed or instantaneous CHECK ...... the speed of an speed, we are talking about the rates at which distance is traveled. object by dividing the distance think! covered by time. If we know average speed and travel time, the distance traveled is easy to find. A simple rearrangement of the definition above gives The speedometer in every car also has an odometer Teaching Resources total distance covered ϭ average speed × travel time that records the distance • Reading and Study traveled. If the odometer Workbook For example, if your average speed is 80 kilometers per hour on a reads zero at the begin- 4-hour trip, then you cover a total distance of 320 kilometers. ning of a trip and 35 km • Laboratory Manual 8 a half hour later, what is • PresentationEXPRESS CONCEPT How can you calculate speed? the average speed? • Interactive Textbook
CHECK ...... Answer: 4.2.2 • Next-Time Questions 4-1, 4-2, 4-3
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Key Term 4.3 Velocity velocity Velocity is directed In everyday language, we can use the words speed and velocity inter- Common Misconception speed. changeably. In physics, we make a distinction between the two. Very Speed and velocity are the same. simply, the difference is that velocity is speed in a given direction. FACT Speed is the rate at which When we say a car travels at 60 km/h, we are specifying its speed. something moves; velocity is But if we say a car moves at 60 km/h to the north, we are specifying speed in a given direction. its velocity. Speed is a description of how fast an object moves; Teaching Tip Define velocity velocity is how fast and in what direction it moves. (speed with direction). Tell your A quantity such as velocity that specifies direction as well as mag- students that the distinction nitude is called a vector quantity. Recall in Chapter 2 that quantities between speed and velocity will that require only magnitude for a description are scalar quantities. be covered later with vectors in Chapter 5. Point out the more Speed is a scalar quantity. Velocity, like force, is a vector quantity. important distinction between average speed or velocity and Constant Velocity Constant speed means steady speed. instantaneous speed or velocity. Something with constant speed doesn’t speed up or slow down. It is the average speed for a Constant velocity, on the other hand, means both constant speed and particular trip that is usually constant direction. Constant direction is a straight line—the object’s calculated. path doesn’t curve. So constant velocity means motion in a straight Teaching Tip Explain line at constant speed. The car in Figure 4.5 may have a constant that a car’s speed gauge is speed, but its velocity is changing. called a speedometer and not a velocimeter because the FIGURE 4.5 speedometer of a car measures the speed at any instant but not The car on the circular its direction. track may have a constant speed but not a constant Ask If an airplane travels velocity, because its direc- 500 km due north in 1 h, what is tion of motion is changing its velocity? If it travels 250 km every instant. due north in 1/2 h? 125 km north in 1/4 h? Each answer is the same—500 km/h north.
think! Changing Velocity If either the speed or the direction (or both) is changing, then the velocity is changing. Constant speed and constant The speedometer of a velocity are not the same. A body may move at constant speed along car moving northward CONCEPT Speed is a description a curved path, for example, but it does not move with constant veloc-
CHECK ...... of how fast an object reads 60 km/h. It passes moves; velocity is how fast and in another car that travels ity, because its direction is changing every instant. what direction it moves. southward at 60 km/h. In a car there are three controls that are used to change the veloc- Do both cars have the ity. One is the gas pedal, which is used to maintain or increase the same speed? Do they Teaching Resources speed. The second is the brake, which is used to decrease the speed. have the same velocity? The third is the steering wheel, which is used to change the direction. • Reading and Study Answer: 4.3 Workbook CONCEPT How is velocity different from speed?
• Laboratory Manual 9 CHECK ...... • PresentationEXPRESS • Interactive Textbook
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4.4 Acceleration think! Key Term We can change the state of motion of an object by changing its Suppose a car moving acceleration speed, its direction of motion, or both. Any of these changes is a in a straight line steadily increases its speed each Common Misconception change in velocity. Sometimes we are interested in how fast the second, first from 35 to Acceleration is simply an increase in velocity is changing. A driver on a two-lane road who wants to 40 km/h, then from 40 speed. pass another car would like to be able to speed up and pass in the to 45 km/h, then from FACT Acceleration is any change shortest possible time. Acceleration is the rate at which the velocity 45 to 50 km/h. What is in speed, direction, or both. 4.4.1 is changing. You can calculate the acceleration of an object by its acceleration? Teaching Tip Write the dividing the change in its velocity by time. Answer: 4.4.1 equation for acceleration on the board and emphasize the word change of velocity acceleration change. Explain that it is the time interval change in motion that one feels in a vehicle when accelerating. We are familiar with acceleration in an automobile, such as the Can you see that the We lurch in a vehicle when it one shown in Figure 4.6. The driver depresses the gas pedal, which is gas pedal (accelerator), undergoes changes in speed or appropriately called the accelerator. The passengers then experience brakes, and steering direction. State there are three acceleration, or “pickup” as it is sometimes called, as they are pressed wheel in an automobile controls in an automobile that into their seats. The key idea that defines acceleration is change. are all controls for make the vehicle accelerate—the acceleration? accelerator pedal, the brakes, Whenever we change our state of motion, we are accelerating. A car and the steering wheel. that can accelerate well has the ability to change its velocity rapidly. Teaching Tip State that A car that can go from zero to 60 km/h in 5 seconds has a greater we accelerate on a curve even acceleration than another car that can go from zero to 80 km/h in though our speed may not 10 seconds. So having good acceleration means being able to change change. The direction changes. velocity quickly and does not necessarily refer to how fast something The definition of velocity is moving. includes direction and makes the definition of acceleration all- In physics, the term acceleration applies to decreases as well as encompassing. increases in speed. The brakes of a car can produce large retarding Teaching Tip Give numerical accelerations, that is, they can produce a large decrease per second in examples of acceleration in the speed. This is often called deceleration. We experience deceleration units of kilometers per hour per when the driver of a bus or car slams on the brakes and we tend to second to establish the idea of hurtle forward. acceleration. Be sure that your students are working on the examples with you. For example, ask them to find the acceleration of a car that goes from rest to 100 km/h in 10 s. Do not use examples involving seconds squared until your students taste success with the easier km/h?s examples. Have students check their work with their neighbors as you go along. Only after they FIGURE 4.6 get the hang of it should you A car is accelerating when- introduce examples involving ever there is a change in its m/s?s and develop a sense for 2 state of motion. customary units m/s .
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Ask What is the acceleration direction of your velocity, of a car that goes from rest to you speed up; against your 100 km/h in 10 s? 10 km/h?s velocity, you slow down; at What is the acceleration of a an angle to your velocity, mechanical part that moves your direction changes. from 0 to 10 m/s in a time of 1 s? 10 m/s2
Change in Speed When straight-line motion is considered, it is common to use speed and velocity interchangeably. When the direc- tion is not changing, acceleration may be expressed as the rate at which speed changes. change in speed acceleration (along a straight line) time interval Speed and velocity are measured in units of distance per time. CONCEPT You can calculate the Since acceleration is the change in velocity or speed per time interval,
CHECK ...... acceleration of an its units are those of speed per time. If we speed up, without chang- object by dividing the change in think! ing direction, from zero to 10 km/h in 1 second, our change in speed its velocity by time. In 5 seconds a car moving is 10 km/h in a time interval of 1 s. Our acceleration along a straight in a straight line increases line is Teaching Resources its speed from 50 km/h change in speed 10 km/h acceleration 10 km/h s • Reading and Study to 65 km/h, while a truck time interval 1 s Workbook goes from rest to 15 km/h in a straight line. Which The acceleration is 10 km/h.s, which is read as “10 kilometers • Problem-Solving Exercises 4.4.2. in Physics 2-1 undergoes greater per hour-second.” Note that a unit for time appears twice: once acceleration? What is • Laboratory Manual 11 for the unit of speed and again for the interval of time in which the the acceleration of speed is changing. • Probeware Lab Manual 1 each vehicle? • PresentationEXPRESS CONCEPT Answer: 4.4.2 How do you calculate acceleration?
• Interactive Textbook CHECK ......
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FIGURE 4.8 If a falling rock were somehow equipped with a speedometer, in each succeeding second of fall its reading would increase by the same amount, 10 m/s. Table 4.2 shows the speed we would read at various seconds of fall.
In Figure 4.8 we imagine a freely-falling boulder with a speed- ometer attached. As the boulder falls, the speedometer shows that the boulder acquires 10 m/s of speed each second. This 10 m/s gain each second is the boulder’s acceleration.
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discover! MATERIALS dollar bill Can You Catch a Falling Bill? EXPECTED OUTCOME It will be impossible for the friend to 1. Have a friend hold a dollar bill so catch the falling dollar bill the midpoint hangs between your because the reaction time is fingers. greater than the time it takes 2. Have your friend release the bill the bill to fall 8 cm. without warning. Try to catch it! THINK Less than 1/8 of a 3. Think How much reaction time do you have when second to catch the bill your friend drops the bill?