Chapter 7 Problems
7–1 and 7–2 Momentum
and Its Conservation
1.
(I) What is the magnitude of the momentum of a 28-g sparrow flying with a
speed of
2.4 m/s?
2.
(I) A constant friction force of 25 N acts on a 65-kg skier for 20 s. What
is the skier’s change in velocity?
3. (II) A 0.145-kg baseball pitched at 39.0 m/s is hit on a horizontal line drive straight back toward the pitcher at 52.0 m/s.
If the contact time between bat and ball is 3.00 x 10-3 s.
Calculate the average force between the ball and bat during contact.
4. (II) A child in a boat throws a 6.40-kg package out horizontally with a speed of 10.0 m/s as shown in the figure below
Calculate the velocity of the boat immediately after, assuming it was initially at rest.
The mass of the child is 26.0 kg, and that of the boat is 45.0 kg. Ignore
water resistance.
5. (II) Calculate the force exerted on a rocket, given that the propelling gases are expelled at a rate of 1500 kg/s with a speed
of
4.0 x104
m/s (at
the moment
of takeoff).
6. (II) A 95-kg halfback moving at 4.1 m/s on an apparent breakaway for a touchdown is tackled from behind.
When he was tackled by an 85-kg cornerback running at 3.5 m/s in the same direction, what was their mutual speed
immediately after the tackle?
7. (II) A 12,600-kg railroad car travels alone on a level frictionless track with a constant speed of 18.0 m/s.
A 5350-kg load, initially at rest, is dropped onto the car.
What will be the car’s new speed?
8. (II) A 9300-kg boxcar traveling at 15.0 m/s strikes a second boxcar at rest.
The two stick together and move off
with a speed of 6.0 m/s. What is the mass of
the second car?
9. (II) During a Chicago storm, winds can whip horizontally at speeds of 100 km/h.
If the air strikes a person at the rate of 40 kg/s per square meter and is brought to rest, estimate the force of the wind
on a person. Assume the person is .50 m high and 0.50 m wide.
Compare to the typical maximum force of friction μ about 1.0 between the person and the ground, if the person has a
mass of 70 kg.
10. (II) A 3800-kg open railroad car coasts along with a constant speed of 8.60 m/s on a level track.
Snow begins to fall vertically and fills the car at a rate of 3.50 kg/min. Ignoring friction with the tracks, what is the speed of
the car after 90.0
min?
11. (II) An atomic nucleus initially moving at 420 m/s emits an alpha particle in the direction of its velocity, and the remaining
nucleus slows to 350 m/s.
If the alpha particle has a mass of 4.0 u and the original nucleus has a mass of 222 u, what speed does the alpha particle
have when it is emitted?
12. (II) A 23-g bullet traveling 230 m/s penetrates a 2.0-kg block of wood and emerges cleanly 170 m/s.
If the block is stationary on a frictionless surface when hit, how fast does
it move after the bullet emerges?
13. (III) A 975-kg two-stage rocket is traveling at a speed of 5.80 x103 m/s with respect to Earth when a pre-designed explosion
separates the rocket into two sections of equal mass that then
move at a speed of 2.20 x103 m/s relative to each other along the original line of motion.
(a) What are the speed and direction of each section (relative to Earth) after the explosion?
(b) How much energy was supplied
by the explosion? [Hint: What is
the change in ke as a result
of the explosion?]
14. (III) A rocket of total mass 3180 kg is traveling in outer space with a velocity of 115 m/s.
To alter its course by 35.0º, its rockets can be fired briefly in a direction perpendicular to its original motion.
If the rocket gases are expelled at a speed of 1730 m/s, how much mass must
be expelled?
15. (II) A golf ball of mass 0.045 kg is hit off the tee at a speed of 45 m/s. The golf club was in contact with the ball for
3.5 x10-3 s. Find
(a) the impulse imparted to the golf ball, and
(b) the average force exerted on
the ball by the golf club.
16. (II) A 12-kg hammer strikes a nail at a velocity of 8.5 m/s and comes to rest in a time interval of 8.0 ms.
(a) What is the impulse given to the nail?
(b) What is the average force
acting on the nail?
17. (II) A tennis ball of mass m = 0.060 kg and speed = 25 m/s strikes a wall at a 45º angle and rebounds with the same
speed at 45º (Fig. below). Another figure
What is the impulse (magnitude and direction) given to the ball?
18. (II) You are the design engineer in charge of the crashworthiness of new automobile models. Cars are tested by smashing
them into fixed, massive barriers at 50 km/h (30 mph). A new model of mass 1500 kg takes 0.15 s from the time of impact
until it is brought to rest.
(a) Calculate the average force exerted on the car by the barrier.
(b)
Calculate the average deceleration of the car.
19.
(II) A 95-kg fullback is running at
to the east and is
stopped in 0.75 s by a head-on tackle by a tackler running due
west. Calculate
(a) the original momentum of the fullback,
(b) the impulse exerted on the fullback,
(c) the impulse exerted on the tackler, and
(d)
the average force exerted on the tackler.
20.
(II) Suppose the force acting on a tennis ball (mass 0.060 kg) points in the
direction and is given
by the graph of
the figure below as a function of time.
Use graphical methods to estimate
(a) the total impulse given the ball, and
(b) the velocity of the ball after being struck,
assuming the ball is being served
so it is nearly at rest initially.
21. (III) From what maximum height can a 75-kg person jump without breaking the lower leg bone of either leg?
Ignore air resistance and assume the cm of the person moves a distance of 0.60 m from the standing to the seated
position (that is, in breaking the fall).
Assume the breaking strength (force per unit area) of bone is 170x106N/m2 and its smallest cross-sectional area is
2.5x10-4m2
7–4 and 7–5 Elastic
Collisions
22. (II) A ball of mass 0.440 kg moving east ( +x direction)ion) with a speed of 3.30 m/s collides head-on with a 0.220-kg ball
at rest.
If the collision is perfectly elastic, what will be
the speed and
direction of each ball after the collision?
23. (II) A 0.450-kg ice puck, moving east with a speed of 3.00 m/s, has a head-on collision with a 0.900-kg puck initially at
rest. Assuming a perfectly
elastic collision, what will be the speed and direction
of each object after
the collision?
24. (II) Two billiard balls of equal mass undergo a perfectly elastic head-on collision. If one ball’s initial speed was 2.00 m/s and the other’s
was
3.00 m/s in the opposite
direction, what will be their speeds
after the collision?
25. (II) A 0.060-kg tennis ball, moving with a speed o 2.50 m/s collides head-on with a 0.090-kg ball initially moving away from it at a
speed of
1.15 m/s. Assuming a perfectly
elastic collision, what are the speed and direction of each ball after the
collision?
26. (II) A softball of mass 0.220 kg that is moving with a speed of 8.5 m/s collides head-on and elastically with another ball initially at rest.
Afterward the incoming softball bounces backward with a speed of 3.7 m/s. Calculate
(a) the velocity of the target ball after the collision, and
(b)
the mass of the target ball.
27. (II) Two bumper cars in an amusement park ride collide elastically as one approaches the other directly from the rear (Fig. 7–34).
Car A has a mass of 450 kg and car B 550 kg, owing to differences in passenger mass.
If car A approaches at 4.50 m/s and car B is moving at 3.70 m/s calculate
(a) their velocities after the collision, and
(b)
the change in momentum of each.
28. (II) A 0.280-kg croquet ball makes an elastic head-on collision with a second ball initially at rest. The second ball moves off with half the
original speed of the first ball.
(a) What is the mass of the second ball?
(b)
What fraction of the original kinetic energy
(delta KE/KE) gets transferred to
the second ball?
29. (III) In a physics lab, a cube slides down a frictionless incline as shown in Fig. 7–35, and elastically strikes another cube at the bottom
that is only one-half its mass. If the incline is 30 cm high and the table is 90 cm off the floor, where does each cube land?
[Hint:
Both leave the incline moving horizontally.]
30.
(III) Take the general case of an object of mass
and velocity
elastically striking a
stationary
object of mass
head-on.
(a)
Show that the final velocities
and
are given by
(b)
What happens in the extreme case when
is much smaller than
Cite a common example
of this. (c) What happens in the
extreme case when
is much larger than
Cite a common example
of this. (d) What happens in the
case when
Cite a common example.
31. (I) In a ballistic pendulum experiment, projectile 1 results in a maximum height h of the pendulum equal to 2.6 cm.
A second projectile causes the the pendulum to swing twice as high, h2 = 5.2 cm.
The second projectile
was how many times faster than the first?
32. (II) A 28-g rifle bullet traveling 2.30 m/s buries itself in a 3.6-kg pendulum hanging on a 2.8-m-long string, which makes the
pendulum
swing upward in an arc. Determine the vertical and horizontal components of
the pendulum’s displacement.
33. (II) (a) Derive a formula for the fraction of kinetic energy lost, for the ballistic pendulum collision of Example 7–10.
(b) Evaluate for m = 14.0 g NS M = 38 G.
34.
(II) An internal explosion breaks an object, initially at rest, into two
pieces, one of which has 1.5 times the mass of the other. If 7500 J
were
released in the explosion, how much kinetic energy did each piece acquire?
35.
(II) A 920-kg sports car collides into the rear end of a 2300-kg SUV stopped
at a red light. The bumpers lock, the brakes are locked, and
the two cars
skid forward 2.8 m before stopping. The police officer, knowing that the
coefficient of kinetic friction between tires and road
is 0.80, calculates
the speed of the sports car at impact. What was that speed?
36.
(II) A ball is dropped from a height of 1.50 m and rebounds to a height of
1.20 m. Approximately how many rebounds will the ball
make before losing 90%
of its energy?
37.
(II) A measure of inelasticity in a head-on collision of two objects is the
coefficient of restitution, e,
defined as
where
is the relative
velocity of the two objects after the collision and
is their relative
velocity before it. (a) Show that
for a perfectly
elastic collision, and
for a completely
inelastic collision. (b) A simple
method for measuring the coefficient of restitution for an object colliding
with a very hard surface like steel is to drop the object onto a heavy steel
plate, as shown in Fig. 7–36. Determine a formula for
e in terms of the original height
h and the maximum height
reached after one
collision.
38.
(II) A wooden block is cut into two pieces, one with three times the mass of
the other. A depression is made in both faces of the cut, so that a
firecracker can be placed in it with the block reassembled. The reassembled
block is set on a rough-surfaced table, and the fuse is lit. When the
firecracker explodes, the two blocks separate and slide apart. What is the
ratio of distances each block travels?
39.
(III) A 15.0-kg object moving in the
direction at
collides head-on with
a 10.0-kg object moving in the
direction at
Find the final velocity of each mass if:
(a) the objects stick together;
(b) the collision is elastic;
(c) the 15.0-kg object is at rest after the collision;
(d) the 10.0-kg object is at rest after the collision;
(e)
the 15.0-kg object has a velocity of
in the
direction after the
collision. Are the results in (c),
(d), and (e)
“reasonable”? Explain.
46. (I) Find the center of mass of the three-mass system shown in the figure below.
Specify relative to the left-hand 1.00-kg mass.
47. (I) The distance between a carbon atom (MC = 12 and an oxygen atom (MO = 16 μ) in the CO molecule
is
1.13X10-10
M.
How far from the
carbon atom is the center of mass of the molecule?
48. (I) The cm of an empty 1050-kg car is 2.50 m behind the front of the car.
How far from the front of the car will the cm be when two people sit in the front seat 2.80 m from the front of the car,
and three people sit
in the back seat 3.90 m from the front?
Assume that each person has a mass
of 70.0 kg.
49. (II) A square uniform raft, 18 m by 18 m, of mass 6800 kg, is used as a ferryboat. If three cars, each of mass
1200 kg, occupy its NE,
SE, and SW
corners, determine the cm of
the loaded ferryboat.
50.
(II) Three cubes, of sides
l0,
2l0
and 3l0
are
placed next to one
another (in contact) with their centers along a straight line and the
l = 2l0
cube in the center
(Fig. 7–39). What is the position, along this line, of the
cm of this system? Assume the
cubes are made of the
same uniform material.
51. (II) A (lightweight) pallet has a load of identical cases of tomato paste (see Fig. 7–40), each of which is a cube of length l.
Find the center of gravity in
the horizontal plane, so that the crane operator can pick up the load
without tipping it.
52.
(III) A uniform circular plate of radius 2R
has a circular hole of radius R
cut out of it. The center
of the smaller circle
is a distance
0.80R from the
center C of the larger circle, Fig. 7–41. What is the position of the center
of mass of the plate? [Hint: Try
subtraction.]