dashed line) represents the displacement. As shown, this area consists of a rectangle and a triangle. Compute their areas and compare the sum of the two areas with the ex-pression on the right-hand side of Equation 2.12.
65.Setting a new world record in a 100-m race, Maggie and Judy cross the finish line in a dead heat, both taking 10.2 s.
Accelerating uniformly, Maggie took 2.00 s and Judy 3.00 s to attain maximum speed, which they maintained for the rest of the race. (a) What was the acceleration of each sprinter? (b) What were their respective maximum speeds?
(c) Which sprinter was ahead at the 6.00-s mark, and by how much?
66.A commuter train travels between two downtown stations.
Because the stations are only 1.00 km apart, the train never reaches its maximum possible cruising speed. Dur-ing rush hour the engineer minimizes the time interval $t between two stations by accelerating for a time interval $t1 at a rate a1!0.100 m/s2 and then immediately braking with acceleration a2! #0.500 m/s2 for a time interval
$t2. Find the minimum time interval of travel $t and the time interval $t1.
67.A hard rubber ball, released at chest height, falls to the pavement and bounces back to nearly the same height.
When it is in contact with the pavement, the lower side of the ball is temporarily flattened. Suppose that the maxi-mum depth of the dent is on the order of 1 cm. Compute an order-of-magnitude estimate for the maximum acceler-ation of the ball while it is in contact with the pavement.
State your assumptions, the quantities you estimate, and the values you estimate for them.
68. At NASA’s John H. Glenn research center in Cleveland, Ohio, free-fall research is performed by dropping experi-ment packages from the top of an evacuated shaft 145 m high. Free fall imitates the so-called microgravity environ-ment of a satellite in orbit. (a) What is the maximum time interval for free fall if an experiment package were to fall the entire 145 m? (b) Actual NASA specifications allow for a 5.18-s drop time interval. How far do the packages drop and (c) what is their speed at 5.18 s? (d) What constant accelera-tion would be required to stop an experiment package in the distance remaining in the shaft after its 5.18-s fall?
An inquisitive physics student and mountain climber climbs a 50.0-m cliff that overhangs a calm pool of water.
He throws two stones vertically downward, 1.00 s apart, and observes that they cause a single splash. The first stone has an initial speed of 2.00 m/s. (a) How long after release of the first stone do the two stones hit the water? (b) What initial velocity must the second stone have if they are to hit simultaneously? (c) What is the speed of each stone at the instant the two hit the water?
70.A rock is dropped from rest into a well. The well is not re-ally 16 seconds deep, as in Figure P2.70. (a) The sound of the splash is actually heard 2.40 s after the rock is released from rest. How far below the top of the well is the surface of the water? The speed of sound in air (at the ambient temperature) is 336 m/s. (b) What If? If the travel time for the sound is neglected, what percentage error is intro-duced when the depth of the well is calculated?
71.To protect his food from hungry bears, a boy scout raises his food pack with a rope that is thrown over a tree limb at height h above his hands. He walks away from the vertical rope with constant velocity vboy, holding the free end of the rope in his hands (Fig. P2.71). (a) Show that the speed v of the food pack is given by x(x2%h2)#1/2vboywhere x
Figure P2.59 (a) The Acela—1 171 000 lb of cold steel thundering along at 150 mi/h. (b) Velocity-versus-time graph for the Acela.
Courtesy Amtrak Nec Media Relations
Time (s) Height (m) Time (s) Height (m)
0.00 5.00 2.75 7.62
0.25 5.75 3.00 7.25
0.50 6.40 3.25 6.77
0.75 6.94 3.50 6.20
1.00 7.38 3.75 5.52
1.25 7.72 4.00 4.73
1.50 7.96 4.25 3.85
1.75 8.10 4.50 2.86
2.00 8.13 4.75 1.77
2.25 8.07 5.00 0.58
2.50 7.90
Height of a Rock versus Time
Table P2.74
is the distance he has walked away from the vertical rope.
(b) Show that the acceleration a of the food pack is h2(x2%h2)#3/2v2boy. (c) What values do the acceleration a and velocity v have shortly after he leaves the point under the pack (x ! 0)? (d) What values do the pack’s velocity and acceleration approach as the distance x continues to increase?
72.In Problem 71, let the height h equal 6.00 m and the speed vboy equal 2.00 m/s. Assume that the food pack starts from rest. (a) Tabulate and graph the speed–time graph.
(b) Tabulate and graph the acceleration-time graph. Let the range of time be from 0 s to 5.00 s and the time inter-vals be 0.500 s.
Kathy Kool buys a sports car that can accelerate at the rate of 4.90 m/s2. She decides to test the car by racing with an-other speedster, Stan Speedy. Both start from rest, but ex-perienced Stan leaves the starting line 1.00 s before Kathy.
If Stan moves with a constant acceleration of 3.50 m/s2 and Kathy maintains an acceleration of 4.90 m/s2, find (a) the time at which Kathy overtakes Stan, (b) the dis-tance she travels before she catches him, and (c) the speeds of both cars at the instant she overtakes him.
73.
74. Astronauts on a distant planet toss a rock into the air.
With the aid of a camera that takes pictures at a steady rate, they record the height of the rock as a function of time as given in Table P2.74. (a) Find the average velocity of the rock in the time interval between each measure-ment and the next. (b) Using these average velocities to approximate instantaneous velocities at the midpoints of the time intervals, make a graph of velocity as a function of time. Does the rock move with constant acceleration? If so, plot a straight line of best fit on the graph and calculate its slope to find the acceleration.
Two objects, A and B, are connected by a rigid rod that has a length L. The objects slide along perpendicular guide rails, as shown in Figure P2.75. If A slides to the left with a constant speed v, find the velocity of B when . ! 60.0°.
75.
m h
v a
x
vboy
α L
y
x
v A B
O x y Figure P2.70
Figure P2.71 Problems 71 and 72.
Figure P2.75 By permission of John Hart and Creators Syndicate, Inc.
Answers to Quick Quizzes
2.1 (c). If the particle moves along a line without changing di-rection, the displacement and distance traveled over any time interval will be the same. As a result, the magnitude of the average velocity and the average speed will be the same. If the particle reverses direction, however, the dis-placement will be less than the distance traveled. In turn, the magnitude of the average velocity will be smaller than the average speed.
2.2 (b). If the car is slowing down, a force must be pulling in the direction opposite to its velocity.
2.3 False. Your graph should look something like the follow-ing. This vx-t graph shows that the maximum speed is about 5.0 m/s, which is 18 km/h (! 11 mi/h), and so the driver was not speeding.
2.4 (c). If a particle with constant acceleration stops and its ac-celeration remains constant, it must begin to move again in the opposite direction. If it did not, the acceleration would change from its original constant value to zero.
Choice (a) is not correct because the direction of accelera-tion is not specified by the direcaccelera-tion of the velocity. Choice (b) is also not correct by counterexample—a car moving in the # x direction and slowing down has a positive accel-eration.
2.5 Graph (a) has a constant slope, indicating a constant accel-eration; this is represented by graph (e).
Graph (b) represents a speed that is increasing constantly but not at a uniform rate. Thus, the acceleration must be increasing, and the graph that best indicates this is (d).
Graph (c) depicts a velocity that first increases at a constant rate, indicating constant acceleration. Then the velocity stops increasing and becomes constant, indicat-ing zero acceleration. The best match to this situation is graph (f).
2.6 (e). For the entire time interval that the ball is in free fall, the acceleration is that due to gravity.
2.7 (d). While the ball is rising, it is slowing down. After reach-ing the highest point, the ball begins to fall and its speed increases.
2.8 (a). At the highest point, the ball is momentarily at rest, but still accelerating at # g.
vx(m/s)
t(s) 6.0
4.0 2.0 0.0 –2.0 –4.0 –6.0
20 30 40 50
10
Answer to Quick Quiz 2.3
Chapter 3
Vectors
C H A P T E R O U T L I N E