Chapter 20 Energy Transferred Thermally Chapter 21 Degradation of Energy Chapter 22 Electric Interactions Chapter 23 The Electric Field Chapter 25 Work and Energy in Electrostatics Chapter 26 Charge Separation and Storage Chapter 27 Magnetic Interactions Chapter 29 Changing Magnetic Fields Chapter 30 Changing Electric Fields Chapter 31 Electric Circuits Chapter 32 Electronics Chapter 33 Ray Optics Chapter 34 Wave and Particle Optics Under what conditions is the average velocity over a time interval equal to the instantaneous velocity at every instant in the interval?

What mathematical relationship allows you to compute the X component of an object's velocity at some instant, given the object's X component of position as a function of time? What is the mathematical meaning of the symbol A, the Greek capital letter delta? The height of a story apartment building D 2. The distance light travels during a human life span B, N 3. The displacement from your mouth of an indigestible popcorn kernel as it passes through your body, and the distance traveled by the same kernel F, O 4.

The time interval within which a batter must react to a fast pitch before it reaches home plate in professional baseball C, H 5. The distance traveled when you nod off for 2 s while driving on the freeway K 7. The time interval for a nonstop flight halfway around the world from Paris, France, to Auckland, New Zealand J and item 7 above The number of revolutions made by a typical car's tires in one year L, E The maximum speed of your right foot while walking A, M , P What is your average walking speed?

What is the speed of light? What is the speed of a fastball thrown by a professional pitcher? What is the height of each story in an apartment building? What distance does a typical car travel during one year? When you are sitting upright, how far above the chair seat is your mouth? What is the distance from the pitchers mound to home plate? What thickness of rubber is lost during the lifetime of a car tire?

What is the circumference of Earth? What is a typical freeway speed? What is the circumference of a car tire? For what time interval is your right foot at rest i f you walk for 2 min? What is a typical human life span? What is the length of the digestive tract in an adult person? If you walk 10 m in a straight line, what is the displacement of your right foot? How many miles of service does a car tire provide? How many revolutions does a car tire make in traveling 1 m? To visualize the motion, we make a sketch that represents the velocity in each of the four segments Figure WG2.

Thus tj in our general kinematic formulas has different values in the different segments and is not always equal to zero. The initial position for each successive segment is equal to the final position for the previous segment. We can connect the start and end positions in any segment with a straight line constant slope because the velocity is constant in each segment.

For motion at constant velocity, we can use Eq. Let us say that all the motion is along an x axis. In addition, we need to determine his average velocity during two time intervals, the distance traveled in 44 s, and the average speed in the s interval. We summarize the given information to see which quantities we know.

We need unambiguous symbols for the quantities in the various segments. Although it would be tempting to label the initial instant rather thah to, we recognize that there are several segments to this m o tion and that the initial instant of motion in one segment is the final For part c, we get the distance traveled in 44 s by adding the distances traveled in the four segments, noting that distance traveled is a magnitude. For part d, the average speed is the distance traveled in 44 s divided by that time interval.

Of our four tasks, the only "hard" part is getting the displacement in each time interval for our position-versus-time graph. To use Eq. Substituting these values into Eq. This is covering a distance that is a little shorter than the length of a football field in a time interval somewhat less than a minute, which is within the realm of possibility for a person pressed for time.

The signs of our position values agree with the positions we drew in Figure WG2. Notice that the average speed of 1. This is to be expected because average speed is based on distance traveled, which is the sum of the magnitudes of the individual displacements, while average velocity is based on displacement, whose magnitude is much less than the distance traveled because of the shopper's backtracking.

Guided Problem 2. There are five traffic lights between your house and the store, and on your trip you reach all five of them just as they change to red.

Draw a diagram that helps you visualize all the driving and stopped segments and your speed in each segment. Does it matter where the traffic lights are located? You start and stop, speed up and slow down. What does the average speed signify in this case, and how is it related to your displacement in each segment? During how long a time interval are you moving? During how long a time interval are you stopped at the lights? What is your displacement for the trip?

How can you apply the answers to questions 3 and 4 to obtain your average velocity? What is the distance traveled, and how is it related to your average speed? How are average velocity and average speed related in this case? Are your answers plausible, and is your result within the range of-your expectations? Runner B starts 2. Runner A a W h o wins the race? I n y o u r calculations, take the race l e ng t h, m , t o be an exact value. Runner A w i n s because she crosses the f i n i s h line first.

T h e i r distance apart w h e n r u n n e r A crosses the finish line is instant, b u t we do n o t k n o w whether this instant is before or after the slower r u n n e r A crosses the m m a r k o n the track. The t i m e intervals o f approximately 12 s t o r u n m are reasonable. We can also solve the second p ar t o f this p r o b l e m i n a different way. Runner A has a 2. That result is the same as the one we obtained above. What is its average velocity for the entire trip?

We are not given the distance between the airports, and so we have to use the variable d to represent this distance, as shown in Figure WG2. The X component ofthe velocity during each segment is equal to the change in position divided by the time interval for that segment.

Knowing this allows us to express the unknown time intervals in terms of the equal but unknown distances. So we have to devise a way to relate these two unknown quantities to the average velocity.

The distance traveled in each segment of the trip is the same, but the segment with the slower speed requires a longer time interval. With this insight, we can draw a position-versus-time graph for the plane's motion Figure WG2.

We include a dashed line to represent a plane that leaves the first airport at the same instant our plane leaves, travels at the average speed of our plane for its whole trip, and arrives at the second airport at the same instant our plane arrives. The slope of this dashed line represents the average velocity that we seek.

The value of the answer is plausible because it lies between the two velocities at which the airplane actually flies. These two time intervals are much shorter than the time interval needed for the flight, and so our simplification is a reasonable one.

However, he sees no hiker when he arrives. Radioing the control tower, he learns that the hiker is actually a distance d west of the airport. Assume the X axis points east and has its origin at the airport, b What is the helicopter's average velocity for the trip? Draw a diagram representing the motion during the trip, indicating the position of the airport, the turning point, and the rendezvous location.

Label the eastward and westward segments with symbols you can use to set up appropriate equations. Then indicate how your chosen symbols relate to the given information, d and v. Add labeled vector arrows to your diagram to indicate the velocity in each segment. Because no numerical values are given, what physical quantities do you expect to be represented by algebraic symbols in your answer?

Be careful about the signs of the x component of the displacement and the x component of the velocity in each segment of the trip. Is Ax in the eastward segment positive or negative? Is i n this segment positive or negative?

What about Ax and f J. How can these quantities be expressed in terms of v and d? What time interval is needed to fly the eastward segment? What time interval for the westward segment? How can you combine this information to determine the X component of the average velocity and then the average velocity in vector form?

Is the sign on what you expect based on the displacement of the helicopter? How does this problem differ from Worked Problem 2. How are the two problems similar? Worked Problem 2.

Equation 2. According to Eq. We can then evaluate at the specified instants to get numerical values. For part b, the x component of the average velocity can be determined using Eq. This is consistent with the fact that the position x at 2. It is also reassuring that the X component of the average velocity computed from the displacement for the interval from 1. Their positions as a function of time are plotted in Figure WG2.

The street runs east-west, and the positive X direction is eastward. Use Figure WG2. Of those who are, is the x component of their velocity positive or negative? Are they speeding up or slowing down? The lowest average speed? Figure WG2. Use the curves in Figure WG2. How can you determine, from gji x t curve, the direction of a child's motion at any instant, and how is that information related to velocity and to speed? What does motion at constant velocity look like on a posiiionversus-time graph?

What does a changing-velocity situation look like on the graph? How can you tell which child has the highest speed and which, the lowest speed? What feature in the graph would indicate that one child has passed another? In other words, what is the relationship of their positions at the instant one passes another? Your class observed several different objects i n m o t i o n along different lines. Figure P2. T h e y have labeled the horizontal 2. W h i c h o f the graphs could be describing the same motion?

The sequence i n Figure P2. M a k e a graph showing the distance f r o m the leading edge o f the ball to the closest part o f the w a l l using the w a l l as the o r i g i n as Figure P2.

T h e sequence i n Figure P2. I n the first frame the ball is released. I n subsequent frames the ball falls, bounces o n the g r o u n d , rises, time and bounces again. W h a t is y o u r distance traveled? W h a t is y o u r displacement? F r o m start to finish, what is the displacement o f the winner? T h e observations are t o be plotted using the horizontal f r o m a f i l m clip o f a m o v i n g object. Y o u d r a w a position-versus-time graph o f the m o t i o n 6 20 For the m o t i o n represented i n Figure P2.

Today you are in a hurry. I f the trip still takes 12 h, what is your average speed in the last third of the distance? What is the position of the cart after 0. Draw a graph a of the x coordinate of the bug's displacement as a function of time and b of the X component of its velocity as a function of time.

You and a friend work in buildings four equal-length blocks apart, and you plan to meet for lunch. Your friend strolls leisurely at 1.

Knowing this, you pick a restaurant between the two buildings at which you and your friend will arrive at the same instant if both of you leave your respective buildings at the same instant. In blocks, how far from your building is the restaurant?

Which object has the greater displacement over the time interval shown in the graph? You and a friend ride bicycles to school. During the trip your friend has a flat tire that takes him 12 min to frx. If the distance to school is 15 km, which of you gets to school first? You are going on a bicycle ride with a friend.

You start 3. As you chat, your friend pedals by, oblivious to you. When you are halfway back, your friend catches up with you, having herself remembered about the basket, a What was the magnitude of her average velocity for the trip up to that instant? You and your roommate are moving to a city m i away. The two of you begin the trip at the same instant. An hour after leaving, you decide to take a short break at a rest stop. If you are planning to arrive at your destination a half hour before your roommate gets there, how long can you stay at the rest stop before resuming your drive?

You are jogging eastward at an average speed of 2. Once you are 2. Main Menu Utility Menu Search. The Principles and Practice of Physics is a groundbreaking new calculus-based introductory physics textbook that uses a unique organization and pedagogy to allow students to develop a true conceptual understanding of physics alongside the quantitative skills needed in the course.

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