We always describe the position and velocity of an object with reference to a par- ticular coordinate system. When we speak of the velocity of a moving car, we usually mean its velocity with respect to an observer who is stationary on the earth. But when two observers measure the velocity of a moving object, they get different results if one observer is moving relative to the other. The velocity seen by a particular observer is called the velocity relativeto that observer, or simply the relative velocity.
Figure 2.29 shows an example of the concept of relative velocity in action. In this example we’ll assume that all the motions are along the same straight line, thex axis. A woman walks in a straight line with a velocity (xcomponent) of
2 2 2 2
2 v
2 2 2 2
䊱FIGURE 2.28 (a) Position and (b) velocity as functions of time for a ball thrown upward with an initial velocity of 15 m/s.
2.7 Relative Velocity along a Straight Line 55
along the aisle of a train that is moving along the same line with a veloc- ity of What is the woman’s velocity? This seems like a simple enough question, but in reality it has no single answer. As seen by a passenger sitting in the train, she is moving at A person on a bicycle standing beside the train
sees the woman moving at An observer in another
train going in the opposite direction would give still another answer. The velocity is different for different observers. We have to specify which observer we mean, and we speak of the velocityrelativeto a particular observer. The woman’s veloc- ity relative to the train is her velocity relative to the cyclist is
and so on. Each observer, equipped in principle with a meterstick and a stop- watch, forms what we call aframe of reference.Thus, a frame of reference is a coordinate system plus a timer.
Let’s generalize this analysis. In Figure 2.29, we call the cyclist’s frame of reference (at rest with respect to the ground) Cand we call the frame of reference of the moving train T(Figure 2.29b). The woman’s position at any time, relative to a reference point in the moving train (i.e, in frame of reference T), is given by the distance Her position at any time, relative to a reference point that is stationary on the ground (i.e, in the cyclist’s frame C), is given by the distance and the position of the train’s reference point relative to the stationary one is We choose the reference points so that at time all these x’s are zero. Then we can see from the figure that at any later time they are related by the equation
(2.15) That is, the total distance from the origin of Cto the woman at point Wis the sum of the distance from the origin of the cyclist’s frame to the origin of the train’s frame, plus the distance from to the position of the woman.
Because Equation 2.15 is true at any instant, it must also be true that the rate of changeof is equal to the sum of the rates of change of and But the rate of change of is just the velocity of Wwith respect to C, which we’ll denote as and similarly for and Thus, we arrive at the velocity relation
(2.16) Remember that Cis the cyclist’s frame of reference,Tis the frame of refer- ence of the train, and point Wrepresents the woman. Using the preceding nota- tion, we have
From Equation 2.16, the woman’s velocity relative to the cyclist on the ground is
as we already knew.
In this example both velocities are toward the right, and we have implicitly taken that as the positive direction. If the woman walks toward the leftrelative to the train, then and her velocity relative to the cyclist is The sum in Equation 2.16 is always an algebraicsum, and any or all of the veloc- ities may be negative.
When the woman looks out the window, the stationary cyclist on the ground appears to her to be moving backward; we can call the cyclist’s velocity relative to the woman on the train Clearly, this is just the negative of In gen- eral, if AandBare any two points or frames of reference,
(2.17) vA/B5 2vB/A.
vW/C. vC/W.
2.0 m/s.
vW/T5 21.0 m/s,
vW/C51.0 m/s13.0 m/s54.0 m/s,
vW/C
vW/T51.0 m/s, vT/C53.0 m/s.
vW/C5vW/T1vT/C. xT/C. xW/T
vW/C,
xW/C
xT/C. xW/T
xW/C
OT
OT OC
xW/C5xW/T1xT/C.
t50 xT/C.
OC
OT xW/C,
OC xW/T.
OT
4.0 m/s,
1.0 m/s,
1.0 m/s13.0 m/s54.0 m/s.
1.0 m/s.
3.0 m/s.
1.0 m/s
䊱FIGURE 2.29 (a) A woman walks in a moving train while observed by a cyclist on the ground outside. (b) The woman’s position W, at the instant shown, in two frames of ref- erence: that of the train and that of the cyclist 1C2.
1T2 v
PROBLEM-SOLVING STRATEGY 2.2 Relative velocity
Note the order of the double subscripts on the velocities in our discus- sion; always means “the velocity of A(first subscript) relative to B(sec- ond subscript).” These subscripts obey an interesting kind of algebra, as Equation 2.16 shows. If we regard each one as a fraction, then the fraction on the left side is the productof the fractions on the right sides: For a point Prel- ative to frames A and B, This is a handy rule to use when you apply Equation 2.16. If there are threedifferent frames of reference, A,B, and C, we can write immediately
and so on.
vP/A5vP/C1vC/B1vB/A, P/A5 1P/B2 1B/A2.
vA/B
EXAMPLE 2.12 Relative velocity on the highway
Suppose you are driving north on a straight two-lane road at a constant (Figure 2.30). A truck trav- eling at a constant approaches you (in the other lane, fortunately). (a)What is the truck’s veloc- ity relative to you? (b)What is your velocity with respect to the truck?
104 km/h
88 km/h
䊱FIGURE 2.30 Velocities of you and the truck, relative to the ground.
Theory of Relativity
When we derived the relative-velocity relations, we assumed that all the observers use the same time scale. This assumption may seem so obvious that it isn’t even worth mentioning, but it is precisely the point at which Einstein’s special theory of relativity departs from the physics of Galileo and Newton. When any of the speeds approach the speed of light in vacuum, denoted byc, the velocity-addition equation has to be modified. It turns out that if the woman could walk down the aisle at 0.30cand the train could move at 0.90c, then her speed relative to the ground would be not 1.20cbut 0.94c. No material object can travel faster thanc;
we’ll return to the special theory of relativity later in the book.
S O L U T I O N
S E T U P Let you beY, the truck beT, the earth beE, and let the positive direction be north (Figure 2.30). Then
S O LV E Part (a): The truck is approaching you, so it must be
moving south, giving We want to find
Modifying Equation 2.16, we have
The truck is moving south relative to you.
Part (b): From Equation 2.17,
You are moving north relative to the truck.
R E F L E C T How do the relative velocities of Example 2.9 change after you and the truck have passed? They don’t change at all!
The relative positions of the objects don’t matter. The velocity of 192 km/h
vY/T5 2vT/Y5 212192 km/h2 5 1192 km/h.
192 km/h
5 2104 km/h288 km/h5 2192 km/h.
vT/Y5vT/E2vY/E vT/E5vT/Y1vY/E,
vT/Y. vT/E5 2104 km/h.
vY/E5 188 km/h.
the truck relative to you is still but it is now moving away from you instead of toward you.
2192 km/h,
vS
vS
SUMMARY
Displacement and Velocity
(Sections 2.1 and 2.2)When an object moves along a straight line, we describe its position with respect to an origin Oby means of a coordinate such as x. If the object starts at position at time and arrives at position at time the object’s displacementis a vec- tor quantity whose x component is The displace- ment doesn’t depend on the details of how the object travels between and
The rate of change of position with respect to time is given by theaverage velocity,a vector quantity whose xcomponent is
(2.3) On a graph ofxversust, is the slope of the line connecting the starting point and the ending point As the average velocity is calculated for progressively smaller intervals of time the average velocity approaches the instantaneous velocity
Acceleration
(Section 2.3)When the velocity of an object changes with time, we say that the object has an acceleration.Just as velocity describes the rate of change of positionwith time, acceleration is a vector quantity that describes rate of change of velocitywith time. For an object with velocity at time and velocity at time the xcomponent of average acceleration is
(2.5) The average acceleration (xcomponent) between two points can also be found on a graph of versust: is the slope of the line connecting the first point at time and the second point at time As with velocity, when the average acceleration is calculated for smaller and smaller intervals of time the aver- age acceleration approaches theinstantaneous acceleration Motion with Constant Acceleration
(Section 2.4)When an object moves with constant acceleration in a straight line, the following two equations describe its position
and velocity as functions of time: (Equa-
tion 2.12), and (Equation 2.8).
In these equations, and are, respectively, the position and velocity at an initial time and xand are, respectively, the position and velocity at any later time t. The final position xis the sum of three terms: the initial position plus the distance that the body would move if its velocity were constant, plus an addi- tional distance caused by the constant acceleration.
The following equation relates velocity, acceleration, and posi- tion without explicit reference to time:
(Equation 2.13).
vx25v0x212ax1x2x02
1 2at2
v0t x0,
vx t50,
v0x
x0 vx5v0x1axt
x5x01v0xt112axt2 ax. Dt, t2).
(v2x
t1) 1v1x
aav,x vx
aav,x5v2x2v1x
t22t1 5Dv Dt.
t2, v2x
t1 v1x
vx5 lim
DtS0 1Dx/Dt2 1Equation 2.42.
vx: Dt, 1x2,t22.
1x1,t12 vav,x vav,x5x22x1
t22t1 5Dx Dt. x2.
x1
Dx5x22x1. t2,
x2
t1 x1
Continued Summary 57
5 v D 5 2
D 5 2
v 5 5 DD
v v
v 5
5 5Dv D
D 5 2
Dv 5v 2 v
v
v
v v 5v 5v
Proportional Reasoning
(Section 2.5)Many problems have two variables that are related in a simple way. One variable may be a constant times the other, or a con- stant times the square of the other, or a similar simple relationship.
In such problems, a change in the value of one variable is related in a simple way to the corresponding change in the value of the other.
This relation can be expressed as a relation between the quotient of two values of one variable and the quotient of two values of the other, even when the constant proportionality factors are not known.
Freely Falling Objects
(Section 2.6)A freely falling object is an object that moves under the influence of a constant gravitational force. The term free fall includes objects that are initially at rest, as well as objects that have an initial upward or downward velocity. When the effects of air resistance are excluded, all bodies at a particular location fall with the same acceleration, regardless of their size or weight. The con- stant acceleration of freely falling objects is called theacceleration due to gravity,and we denote its magnitude with the letterg. At or near the earth’s surface, the value ofgis approximately
Relative Velocity along a Straight Line
(Section 2.7)When an object Pmoves relative to an object (or refer- ence frame) B, and Bmoves relative to a second reference frame A, we denote the velocity of Prelative to Bby the velocity of P relative to Aby and the velocity of Brelative to Aby These velocities are related by this modification of Equation 2.16:
vP/A5vP/B1vB/A.
vB/A. vP/A,
vP/B,
9.8 m/s2.
Conceptual Questions
1. Give an example or two in which the magnitude of the dis- placement of a moving object is (a) equal to the distance the object travels and (b) less than the distance the object travels.
(c) Can the magnitude of the displacement ever be greater than the distance the object travels?
2. Does the speedometer of a car measure speed or velocity?
3. Under what conditions is average velocity equal to instanta- neous velocity?
4. If an automobile is traveling north, can it have an acceleration toward the south? Under what circumstances?
5. True or false? (a) If an object’s average speed is zero, its aver- age velocity must also be zero. (b) If an object’s average veloc- ity is zero, its average speed must also be zero. Explain the reasoning behind your answers. If the statement is false, give several examples to show that it is false.
6. Is it possible for an object to be accelerating even though it has stopped moving? How? Illustrate your answer with a simple example.
7. Can an object with constant acceleration reverse its direction of travel? Can it reverse its direction twice? In each case, explain your reasoning.
8. Under constant acceleration the average velocity of a particle is half the sum of its initial and final velocities. Is this still true if the acceleration is notconstant? Explain your reasoning.
9. If the graph of the velocity of an object as a function of time is a straight line, what can you find out about the acceleration of this object? Consider cases in which the slope is positive, neg- ative, and zero.
10. If the graph of the position of an object as a function of time is a straight line, what can you find out about the velocity of this object? Consider cases in which the slope is positive, negative, and zero.
11. The following table shows an object’s position xas a function of time t:
6.50 6.75 7.00 7.25 7.50 7.75
2.00 4.00 6.00 8.00 10.0 12.0
(a) Just by looking at the table, what can you conclude about this object’s velocity and acceleration? (b) To see if you are cor- rect, calculate this object’s average velocity and acceleration for a few time intervals. (c) How far did the object move betweent52.00 sandt512.0 s?
t1in s2 x1in m2
Linear Quadratic
y y y
y51/x y51/x2 y5 x2
y5 x
y
x x
x x
Inverse Inverse square Four mathematical relationships common in physics
ay5 2g 5 29.80 m/s2
The motion of an object tossed up and allowed to fall. (For clarity, we show a U-shaped path.) The object is in free fall throughout.
vS
vS vS
vS
vS
vS
vS vS vS 5 1
For instructor-assigned homework, go to www.masteringphysics.com
59 Multiple-Choice Problems 12. The following table shows an object’s velocity vas a function
of time t:
3.40 3.80 4.20 4.60 5.00 5.40
5.00 8.00 11.0 14.0 17.0 20.0
(a) Just by looking at the table, what can you conclude about this object’s acceleration? (b) To see if you are correct, calcu- late the object’s average acceleration for several time intervals.
13. A dripping water faucet steadily releases drops 1.0 s apart. As these drops fall, will the distance between them increase, decrease, or stay the same? Prove your answer.
14. Figure 2.31 shows graphs of the positions of three different moving objects as a function of time. All three graphs pass through points AandB.(a) What can you conclude about the average velocities of these three objects between points Aand B?Why? b) At point B,what characteristics of the motion do the three objects have in common? That is, are they in the same place, do they have the same velocity or speed, and do they have the same acceleration?
t1in s2 v1in cm/s2
16. Figure 2.33 shows the graph of an object’s position xas a function of time t. (a) Does this object ever reverse its direction of motion? If so, where? (b) Does the object ever return to its starting point? (c) Is the velocity of the object constant? (d) Is the object’s speed ever zero?
If so, where? (e) Does the object have any acceleration?
17. Figure 2.34 shows the graph of an object’s velocity as a function of time t.(a) Does this object ever reverse its direction of motion? b) Does the object ever return to its starting point? (c) Is the object’s speed ever zero? If so, where? (d) Does the object have any acceleration?
18. A ball is dropped from rest from the top of a building of height At the same instant, a second ball is projected vertically upward from the ground level, such that it has zero speed when it reaches the top of the building. When the two balls pass each other, which ball has the greater speed, or do they have the same speed? Explain. Where will the two balls be when they are alongside each other: at height above the ground, below this height, or above this height? Explain.
Multiple-Choice Problems
1. Which of the following statements about average speed is cor- rect? (More than one statement may be correct.)
A. The average speed is equal to the magnitude of the average velocity.
B. The average speed can never be greater than the magnitude of the average velocity.
C. The average speed can never be less than the magnitude of the average velocity.
D. If the average speed is zero, then the average velocity must be zero.
E. If the average velocity is zero, then the average speed must be zero.
2. A ball is thrown directly upwards with a velocity of At the end of 4 s, its velocity will be closest to
A. B. C. D.
3. Two objects start at the same place at the same time and move along the same straight line. Figure 2.35 shows the position x as a function of time t for each object. At point A, what must be true about the motion of these objects? (More than one state- ment may be correct.) A. Both have the same speed.
210 m/s.
220 m/s.
230 m/s.
250 m/s.
120 m/s.
h/2
h.
vx
䊱FIGURE 2.31 Question 14.
15. Figure 2.32 shows graphs of the velocities of three different moving objects as a function of time. All three graphs pass through points AandB.(a) What can you conclude about the average accelerations of these three objects between points A andB?Why? (b) At point A,what characteristics of the motion do the three objects have in common? That is, are they in the same place, do they have the same velocity or speed, and do they have the same acceleration?
䊱FIGURE 2.32 Question 15.
䊱FIGURE 2.33 Question 16.
䊱FIGURE 2.34 Question 17.
䊱FIGURE 2.35 Multiple-choice problem 3.
A
B (1)
(2) (3) x
O t
v
x
O t
v
x
O t
A
12. A cat runs in a straight line.
Figure 2.37 shows a graph of the cat’s position as a func- tion of time. Which of the following statements about the cat’s motion must be true?
(There may be more than one correct choice.)
A. At point c, the cat has returned to the place where it started.
B. The cat’s speed is zero at points aandc.
C. The cat’s velocity is zero at point b.
D. The cat reverses the direction of its velocity at point b.
13. A wildebeest is running in a straight line, which we shall call the xaxis, with the posi- tive direction to the right.
Figure 2.38 shows this ani- mal’s velocity as a function of time. Which of the fol- lowing statements about the animal’s motion must be true? (There may be more than one correct choice.) A. Its acceleration is
increasing.
B. Its speed is decreasing from a to b and increasing frombtoc.
C. It is moving to the right between aandc.
D. It is moving to the left between aandband to the right betweenbandc.
14. A brick falling freely from a helicopter drops 40 meters during a certain 1-second time interval. The distance it will fall in the nextsecond is
A. 45 m B. 50 m C. 56 m D. 80 m
15. A bullet is dropped into a river from a very high bridge. At the same time, another bullet is fired from a gun straight down towards the water. If air resistance is negligible, the accelera- tion of the bullets just before they strike the water
A. is greater for the dropped bullet.
B. is greater for the fired bullet.
C. is the same for both bullets.
D. depends on how high they started.
Problems
2.1 Displacement and Average Velocity
1. • An ant is crawling along a straight wire, which we shall call thexaxis, fromAtoBtoCtoD(which overlapsA), as shown in Figure 2.39. Ois the origin. Suppose you take measure- ments and find thatABis 50 cm,BCis 30 cm, andAOis 5 cm.
(a) What is the ant’s position at pointsA, B, C,andD?(b) Find the displacement of the ant and the distance it has moved over each of the following intervals: (i) fromAtoB,(ii) fromBto C,(iii) fromCtoD,and (iv) fromAtoD.
B. Both have the same velocity.
C. Both are at the same position.
D. Both have traveled the same distance.
4. Basedonlyon dimensional analysis, which formulas couldnot be correct? In each case,xis position, is velocity,ais accel- eration, andtis time. (More than one choice may be correct.) A.
B.
C.
D.
5. An object starts from rest and accelerates uniformly. If it moves 2 m during the first second, then, during the first 3 sec- onds, it will move
A. 6 m. B. 9 m. C. 10 m. D. 18 m.
6. If a car moving at 80 mph takes 400 ft to stop with uniform acceleration after its brakes are applied, how far will it take to stop under the same conditions if its initial velocity is 40 mph?
A. 20 ft B. 50 ft C. 100 ft D. 200 ft
7. Figure 2.36 shows the veloc- ity of a jogger as a function of time. What statements must be true about the jog- ger’s motion? (More than one statement may be correct.) A. The jogger’s speed is
increasing.
B. The jogger’s speed is decreasing.
C. The jogger’s acceleration is increasing.
D. The jogger’s acceleration is decreasing.
8. A certain airport runway of length Lallows planes to acceler- ate uniformly from rest to takeoff speed using the full length of the runway. Because of newly designed planes, the takeoff speed must be doubled, again using the full length of the run- way and having the same acceleration as before. In terms of L, what must be the length of the new runway?
A. B. C. D.
9. A ball rolls off a horizontal shelf a height habove the floor and takes 0.5 s to hit. For the ball to take 1.0 s to reach the floor, the shelf’s height above the floor would have to be
A. 2h B. 3h C. D. 4h
10. A frog leaps vertically into the air and encounters no apprecia- ble air resistance. Which statement about the frog’s motion is correct?
A. On the way up its acceleration is upward, and on the way down its acceleration is downward.
B. On the way up and on the way down its acceleration is downward, and at the highest point its accelera- tion is zero.
C. On the way up, on the way down, and at the highest point its acceleration is downward.
D. At the highest point, it reverses the direction of its acceleration.
11. You slam on the brakes of your car in a panic and skid a dis- tanceXon a straight, level road. If you had been traveling half as fast under the same road conditions, you would have skid- ded a distance
A. B. C. X. D. X
"2 X .
2. X
4.
9.8 m/s2
9.8 m/s2
9.8 m/s2
9.8 m/s2
"2h
L2 L/2
2L 4L
v5v012at a5v2/x
x5v011/2at2
v25v0212at
v
䊱FIGURE 2.37 Multiple-choice problem 12.
䊱FIGURE 2.38 Multiple-choice problem 13.
䊱FIGURE 2.36 Multiple-choice problem 7.
D O C
A B
䊱FIGURE 2.39 Problem 1.
v
v
v