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AP Physics 4: Conservation Laws—Linear Motion
A.
Linear Momentum
1. linear momentum: p = mv (kg•m/s)
2. impulse: J = Ft = mv = p
3. conservation of linear momentum
a. momentum of a system remains constant as long
as no outside forces (gravity, spring, friction, push,
or pull) act on the system
b. collision between particles
1. mAvA + mBvB = mAvA’ + mBvB’
Steps
Algebra
FA = -FB
start with Newton's Law
multiply both side by t
FAt = -FBt
mAvA = -mBvB
substitute mv for Ft
mA(vA’ – vA) = -mB(vB’ – vB)
substitute v' – v for v
-mAvA – mBvB = -mAvA’ – mBvB’
collect like v terms
mAvA + mBvB = mAvA’ + mBvB’
multiply by -1
2. two particles collide and stick together
a. inelastic collisions
b. mAvA + mBvB = (mA + mB)v’
Steps
Algebra
mAvA + mBvB = mAvA’ + mBvB’
start with
substitute v' for vA' and vB' mAvA + mBvB = mAv’ + mBv’
mAvA + mBvB = (mA + mB)v’
simplify
3. two particles collide and bounce off
a. elastic collisions
b. difference in velocity is the same after
collision: vA – vB = -(vA’ – vB’)—proof later
c. solving two equations and two unknowns

fill in vA and vB into vA – vB = -(vA' – vB’)

write expression for vA’ in terms of vB'

substitute vA' expression in equation:
mAvA + mBvB = mAvA’ + mBvB’

solve for vB’

solve for vA’ using the expression for vA' above
4. collisions in two dimensions
a. px is conserved independently of py
b. elastic collision
1. mAvAx + mBvBx = mAvAx’ + mBvBx’
2. mAvAy + mBvBy = mAvAy’ + mBvBy’
3. solve two equations & two unknowns
c. inelastic collision
1. mAvAx + mBvBx = (mA + mB)vx'
2. mAvAy + mBvBy = (mA + mB)vy'
3. solve two equations & two unknowns
c. object explodes into two pieces mA and mB
1.
2.
(mA + mB)v = mAvA’ + mBvB’
opposite inelastic collision equation
Name __________________________
B.
Forms of Energy
1. scalar value measured in joules, J = 1 N•m
2. work, W = F||d (J)
a. Only component of F parallel to d does work
F

b.
c.
d.
d
1. F|| = Fcos  W = (Fcos)d
2. include sign W > 0 when F  d 
Wnet > 0 (acceleration), Wnet < 0 (deceleration)
Wnet = 0
1. Fnet = 0 (lift)
2. F  when d  (orbit)
work done by variable force—stretching a spring
1. graph spring force (Fs) vs. position (x)
Fs
slope = k
Area = W
x
2. Fs = kx  slope = Fs/x = k
3. W = Fsx  area = ½bh = ½x(kx) = ½kx2 = W
3. power, P = W/t (W)
a. rate that work is done: Watt, W = J/s
b. P = Fvav, where v is average (W/t = F(d/t) = Fvav)
c. graphing
1. P = slope of W vs. t graph
2. P = area under F vs. v graph
d. kilowatt-hour is a unit of energy
(1KWh = 3.6 x 106 J)
4. mechanical energy
a. work-energy theorem: work done to an object
increases mechanical energy; work done by an
object decreases mechanical energy
b. scalar quantity, like work
c. kinetic energy—energy of motion
1. positive only
2. K = ½mv2 = p2/2m
Steps
Algebra
v2 = v02 + 2ad
start with
assume Ko = 0  vo = 0 v2 = 2ad
ad = ½v2
solve for ad
K = W = Fd
start with
substitute ma for F
K = (ma)d = m(ad)
substitute ½v2 for ad
K = m(½v2)
K = ½mv2
rearrange
mv = p
start with
m2v2 = p2
square both sides
divide both sides by 2m m2v2/2m = p2/2m
substitute K for ½mv2
K = p2/2m
d. potential energy—energy of relative position
1. gravitational potential energy
a. based on arbitrary zero
(usually closest or farthest apart)
b. Ug = mgh (near the Earth's surface)
Steps
Algebra
Ug = W = Fd
start with
substitute mg for F
Ug = (mg)d
substitute h for d
Ug = mgh
c. Ug = -GMm/r (orbiting system)
1. G = 6.67 x 10-11 N•m2/kg2
2. r = distance from center to center
3. Ug = 0 when r is Ug < 0 for all
values of r because positive work is
needed reach Ug = 0
2. spring (elastic) potential energy, Us = ½kx2
a. Us = W to stretch the spring
b. see work by a variable force above
C.
Conservation of Energy
1. work done on object A by a "nonconservative" force
(push or pull, friction) results in the gain in mechanical
energy for object A equal to the loss of energy by the
source of the nonconservative force
2. work done on object A by a "conservative" force (gravity,
spring) results in the change in form of mechanical
energy (U  K) for object A, but no loss in energy
a. conservative forces (Fg and Fs)
1. Fg  d : Ug  K, Fg  d : K  Ug
2. Fs  d : Us  K, Fs  d : K  Us
b. process isn't 100 % efficient
1. friction (W = Ffd) reduces mechanical energy
2. mechanical energy is converted into random
kinetic energy of the object's atoms and the
temperature increases = heat energy—Q
3. total energy is still conserved
3. work done by object A on object B
a. W = mAad (uses up kinetic energy to decelerate)
b. energy loss by object A = energy gain by object B
c. some energy is lost due to friction
4. examples
Process
Energy
Work done to pull a pendulum bob off center W  Ug
1 Transformation release pendulum
Ug  K
Work done by hit a stationary object
KW
Work done to throw a ball into the air
WK
K  Ug
2 Transformation ball rises and falls
Work done by falling ball dents ground
KW
Work done to load a projectile in spring-gun W  Us
3 Transformation release projectile
Us  K
Work done by projectile penetrates target
KW
5. elastic collision formula proof (p and K are conserved)
½mA(vA2 - vA'2) = ½mB(vB'2 - vB2) (vA2 - vA'2) = (vB'2 - vB2)
mA(vA – vA') = mB(vB' – vB)
(vA – vA') = (vB' – vB)
(vA – vA')(vA + vA') = (vB – vB')(vB + vB')  vA - vB = -(vA' - vB')
(vA – vA')
= (vB' – vB)
6. solving conservation of energy problems

determine initial energy of the object, Eo
o if elevated h distance: Ug = mgh
o if accelerated to v velocity: K = ½mv2
o if spring compressed x distance: Us = ½kx2

determine energy added/subtracted due to an external
push or pull: Wp = ±F||d

determine energy removed from the object by friction:
Wf = Ffd = (mgcosd
o d is the distance traveled
o  is the angle of incline (0o for horizontal)

determine resulting energy, E' = Eo ± Wp – Wf

determine d, h, x or v
o if slides a distance d: 0 = Eo ± Wp – mgcosd'
o if elevates a height h: E' = mgh'
o If compresses a spring x: Us: E' = ½kx'2
o if accelerated to velocity v: E' = ½mv'2

general equation (not all terms apply for each problem)
K + Ug + Us ± Wp – Wf = K' + Ug' + Us''
2
½mv + mgh + ½kx2 ± Fpd – Ffd = ½mv'2 + mgh' + ½kx'2
7. solve ballistics problems


M

(vM = 0)
m
vm
bullet collides inelastically with block: mvm = (M + m)v'
block swings or slides (conservation of energy)
o block swings like a pendulum to height h
o K = Ug  ½(M + m)v'2 = (M + m)gh  h = v'2/2g
o block slides a distance d along a rough surface
o K = Wf  ½(M + m)v'2 = (M + m)gd  d = v'2/2g
A. Linear Momentum
Questions 1-15 Briefly explain your answer.
1. An open cart rolls along a frictionless track while it is
raining. As it rolls, what happens to the speed of the cart
as the rain collects in it?
(A) increase
(B) the same
(C) decrease
2.
A small car and a large truck collide head-on and stick
together. Which one has larger momentum change?
(A) car
(B) truck
(C) tie
Questions 3-4 Two boxes, one heavier than the other, are
initially at rest on a horizontal frictionless surface. The
same constant force F acts on each for 1 s.
3. Which box has more momentum after the force acts?
(A) light
(B) heavy
(C) tie
4.
6.
Instead of the identical car, a large truck is heading toward
you. Which option should you take?
(A) hit the truck (B) hit the wall (C) doesn't matter
7.
A small beanbag and a bouncy rubber ball are dropped
from the same height above the floor. The both have the
same mass. Which would hurt more if it hit you on the
head?
(A) beanbag
(B) rubber ball (C) doesn't matter
9.
Questions 11-12 A uranium nucleus (at rest) undergoes fission
and splits into two fragments, one heavy and the other light.
11. Which fragment has the greater momentum?
(A) heavier one (B) lighter one (C) tie
12. Which fragment has the greater speed?
(A) heavier one (B) lighter one (C) tie
13. Alice (100 lbs) and Bill (150 lbs) are standing on slippery
ice and push off of each other. If Alice slides at 6 m/s, what
speed does Bill have?
(A) 2 m/s
(B) 3 m/s
(C) 4 m/s
(D) 6 m/s
Which box has the greater velocity after the force acts?
(A) light
(B) heavy
(C) tie
Questions 5-6 You drive around a curve in a narrow one-way
street at 30 mph when you see an identical car heading
straight toward you at 30 mph. You have two options hit
the car head-on or hit a massive concrete wall head-on.
5. Which option should you take?
(A) hit the car
(B) hit the wall (C) doesn't matter
8.
10. A box slides with initial velocity 10 m/s on a frictionless
surface and collides inelastically with a stationary identical
box. What is the final velocity of the combined boxes?
(A) 0 m/s
(B) 5 m/s
(C) 10 m/s (D) 20 m/s
A bowling ball and a ping-pong ball are rolling toward you
with the same momentum. If you exert the same force to
stop each one, which takes a longer time to bring to rest?
(A) bowling
(B) ping-pong
(C) tie
You tee up a golf ball and drive it down the fairway.
Assume that the collision is elastic. When the ball leaves
the tee, how does its speed compare to the speed of the
golf club before the collision?
(A) faster
(B) the same
(C) slow
14. A cannon sits on a stationary railroad flatcar with a total
mass of 1000 kg. When a 10-kg cannon ball is fired at a
speed of 50 m/s, what is the recoil speed of the flatcar?
(A) 0 m/s
(B) 0.5 m/s (C) 10 m/s (D) 50 m/s
15. If all three collisions below are totally inelastic, which
one(s) will bring the box on the left to a complete halt?
anchored brick wall
I
II
III
mmm2m ½m
v
v ½v
v
2v
(A) I only
(B) II only (C) III only (D) all three
16. What is the momentum of a 0.5-kg ball traveling at 18 m/s?
17. What force is generated by a racket, which strikes a 0.06-kg
tennis ball that reaches a speed of 65 m/s in 0.03 s?
18. A 25-kg child in a stationary 55-kg boat with a 5-kg
package throws the package out horizontally at 8 m/s.
What is the boat and child's resultant velocity?
19. An 85-kg safety running at 5 m/s tackles a 95-kg fullback
traveling at 4 m/s from behind. What is their mutual speed
just after the tackle?
20. A 0.45-kg ice puck, moving east with a speed of 3.0 m/s,
has a head-on elastic collision with a 0.9-kg puck initially at
rest. What are the resulting speeds and directions?
d.
Calculate the percent differences.
Formula
Calculation
vA'
vB'
23. A 0.50-kg softball is traveling at 40 m/s. A bat makes
contact with the ball for 0.025 s, after which, the ball's
velocity is 35 m/s in the opposite direction (v = -75 m/s).
a. Determine the change in the ball's momentum.
21. A 1.0-kg block traveling at 5.0 m/s in the direction of 30o
south of east collides and sticks with a 2.0-kg block
traveling north at 3.0 m/s. Determine
a. The x-component of the resulting velocity, vx'.
b.
The y-component of the resulting velocity, vy'.
c.
The resultant speed.
d.
The resultant direction.
Collision Lab
22. Observe an elastic collision between a swinging 200-g
weight and a golf ball, and compare the actual post-collision
velocities with the theoretical velocities.
a. Collect the following data.
200-g Weight (A)
Golf Ball (B)
string length LA
mass mB
initial angle A
final angle A'
b.
20o
b.
Determine the average force exerted by the bat.

24. Consider two masses, mA = 1 kg and mB = 3 kg, which can
move on a frictionless surface.
a. Mass A, moving east with a speed of 2 m/s, has a
head-on inelastic collision with mass B, moving west
with a speed of 1 m/s. What is the resulting speed
and direction of the combined masses?
b.
Mass A, moving east with a speed of 2 m/s, has a
head-on elastic collision with mass B initially at rest.
What are the resulting speeds and directions?
c.
Mass A, moving 53o north of east at 5 m/s collides and
sticks to mass B moving north at 3 m/s. What is the
resulting speed and direction of the combined masses?
d.
A spring (k = 600 N/m) compressed 0.1 m is placed
between the two stationary masses and then released.
Mass A has a velocity of 12 m/s west. What is the
speed and direction of Mass B?
table height dyB
distance dxB'
Calculate the following from the data.
Formula
Calculation
dyA
vA
dyA'
vA'
tB'
vB'
c.
vA'
vB'
Calculate the following theoretical values.
Formula
Calculation
B. Forms of Energy
Questions 25-50 Briefly explain your answer.
Questions 25-30 Use the following options
(A) W > 0
(B) W = 0
(C) W < 0
25. Work done to hold a 10-kg bowling ball in your arms.
26. Work done to pull a 10-kg box across a rough floor at
constant speed.
27. Work done by a pitcher to throw a 90-mph fast ball.
37. A system of particles has total momentum of zero. Does it
necessarily follow that the total kinetic energy is zero?
(A) yes
(B) no
28. Work done by a catcher to stop a 90-mph fast ball.
29. Work done to whirl a ball in a circle at the end of a string.
30. You lift a 10-kg box a distance of 2 m.
a. Work done by you.
b.
c.
38. Two objects have the same momentum. Do these two
objects also have the same kinetic energy?
(A) yes
(B) no
Question 39-40 Stone A has twice the mass as stone B. They are
dropped from a cliff and reach a point just above ground.
39. What is the speed of stone A compared to stone B?
(A) vA = vB
(B) vA = ½vB
(C) vA = 2vB
(D) vA = 4vB
Work done by gravity.
Total work done.
31. A box is being pulled up a rough incline by a rope. How
many forces are doing work on the box?
(A) 1
(B) 2
(C) 3
(D) 4
32. Which contributes more to the cost of your electric bill each
month, a 1500-Watt hair dryer or a 600-Watt microwave
oven?
(A) hair drier
(B) microwave oven
(C) both contribute equally
(D) depends upon how long each one is on
Questions 33-34 Alice applied 10 N of force over 3 m in 10 s.
Bill applies the same force over the same distance in 20 s.
33. Who did more work?
(A) Alice
(B) Bill
(C) They did the same
34. Who produced the greater power?
(A) Alice
(B) Bill
(C) They did the same
35. Car A has twice the mass of car B, but they both have the
same kinetic energy. How do their speeds compare?
(A) 2vA = vB
(B) 2vA = vB
(C) 4vA = vB
36. A system of particles has total kinetic energy of zero. What
can you say about the total momentum of the system?
(A) p > 0
(B) p = 0
(C) p < 0
40. What is the kinetic energy of stone A compared to stone B?
(A) KA = KB
(B) KA = ½KB
(C) KA = 2KB
(D) KA = 4KB
41. If a car traveling 60 km/hr can brake to a stop within 20 m,
what is its stopping distance if it is traveling 120 km/hr?
(A) 20 m
(B) 40 m
(C) 80 m
(D) 160 m
42. A car starts from rest and accelerates to 30 mph. Later, it
gets on a highway and accelerates to 60 mph. Which takes
more energy, the 0  30 mph or the 30  60 mph?
(A) 0  30 mph (B) 30  60 mph(C) both the same
43. The work Wo accelerates a car for 0 to v. How much work
is needed to accelerate the car from v to 3v?
(A) 2Wo
(B) 3Wo
(C) 8Wo
(D) 9Wo
44. Blocks A and B of mass mA and mB (mA > mB) slide on a
frictionless floor and have the same kinetic energy when
they hit a long rough stretch ( > 0), which slows them
down to a stop. Which one goes farther
(A) A
(B) B
(C) both the same
45. A golfer making a putt gives the ball an initial velocity of vo,
but he has misjudged the putt, and the ball only travels
one-quarter of the distance to the hole. What speed should
he have given the ball?
(A) 2vo
(B) 3vo
(C) 4vo
(D) 8vo
46. Which can never be negative?
(A) W
(B) K
(C) U
47. You and your friend both solve a problem involving a skier
going down a slope, starting from rest. The two of you
have chosen different levels for h = 0 in this problem.
Which of the quantities will you and your friend agree on?
I. Ug
II. Ug
III. K
(A) I only
(B) II only (C) III only (D) II and III
48. Two paths lead to the top of a big hill. A is steep and direct,
while the B is twice as long but less steep. The change in
potential energy on path A compared to path B is
(A) UA < UB
(B) UA = UB
(C) UA > UB
57. How much power is needed to maintain a speed of 25 m/s
against a total friction force of 200 N?
58. How long will it take a 1750-W motor to lift a 285-kg piano
to a sixth-story window 16 m above?
59. A 1000-kg car travels at 30 m/s against 600-N of friction.
a. How much power does the car engine deliver?
b.
The car accelerates at 2 m/s2. How much power does
the car engine deliver now?
c.
The car goes up a 10o incline at 30 m/s. How much
power does the car engine deliver now?
49. How does the work required to stretch a spring 2 cm
compare with the work required to stretch it 1 cm?
(A) W2 = W1
(B) W2 = 2W1
(C) W2 = 4W1
(D) W2 = 8W1
50. A mass attached to a vertical spring causes the spring to
stretch and the mass to move downward. Which is true
about the sign for Us and Ug?
(A) +Us, +Ug
(B) +Us, –Ug
(C) –Us, +Ug
(D) –Us, –Ug
51. How much work is done to move a 50-kg crate horizontally
10 m against a 150-N force of friction?
52. How much work is done to pull a 100-kg crate horizontally
10 m using a force of 100 N at 30o?
53. How much work is done to carry a 100-kg crate 10 m up a
30o ramp?
54. Why is work not needed to keep the earth orbiting the sun?
60. What is the kinetic energy of a 2-kg block moving at 9 m/s?
61. What is the gravitational potential energy of a 2-kg block
that is 6 m above zero potential energy?
62. What is the gravitational potential energy of the earth-moon
system? (MEarth = 5.97 x 1024 kg, Mmoon = 7.35 x 1022 kg,
distance between Earth and moon, r = 3.84 x 108 m)
63. Consider the following spring (k = 100 N/m).
a. Calculate the force (F = kx) needed to stretch a spring
from 0.0 m to 0.5 m.
x (m)
0.0
0.1
0.2
0.3
0.4
0.5
F (N)
b. Graph the data
40 N
55. How much power is needed to change the speed of a
1500-kg car from 10 m/s to 20 m/s in 5 s?
20 N
56. How much power does a 75-kg person generate when
climbing 50 steps (rise of 25 cm per step) in 12 s?
a. in Watts
b.
in horse power (1hp = 746 W)
0N
c.
d.
0.2 m
Calculate the area under the graph.
0.4 m
How much potential energy is stored in the stretched
spring?
Human Power Lab
64. Run up a flight of stairs at the football stadium and calculate
the power that you generated.
a. Collect the following data.
weight Fg-lbs
Number of steps N
time t
b.
height of step y
Questions 68-69 Alice and Bill start from rest at the same time
on frictionless water slides with different shapes.
 Bill
 Alice
width of step x
Calculate the following from the data.
Formula
Calculation
68. At the bottom whose velocity is greater?
(A) Alice
(B) Bill
(C) tie
dy
dx
69. Who makes it to the bottom first?
(A) Alice
(B) Bill
(C) both are the same
d
v
70. A cart starting from rest rolls down a hill and at the bottom
has a speed of 4 m/s. If the cart were given a push, so its
initial speed at the top of the hill was 3 m/s, what would be
its speed at the bottom?
(A) 4 m/s
(B) 5 m/s
(C) 6 m/s
(D) 7 m/s
Fg
m
71. You see a leaf falling to the ground with constant speed.
When you first notice it, the leaf has initial total mechanical
energy Ei. You watch the leaf until just before it hits the
ground, at which point it has final total mechanical energy
Ef. How do these total energies compare?
(A) Ei < Ef
(B) Ei = Ef
(C) Ei > Ef
K
Ug
P
c.
Calculate the power in horse power (1 hp = 746 W).
C. Conservation of Energy
Questions 65-73 Briefly explain your answer.
65. Three balls of equal mass start from rest and roll down
different ramps. All ramps have the same height. Which
ball has the greatest speed at the bottom of the ramp?
 
A
(A) A
B
(B) B
C
(C) C
(D) All the same
66. A stationary block slides down a frictionless ramp and
attains a speed of 2 m/s. To achieve a speed of 4 m/s, how
many times higher must the ball start from?
(A) 2 times
(B) 4 times
(C) 8 times
72. You throw a ball straight up into the air. In addition to
gravity, the ball feels a force due to air resistance.
Compared to the time it takes the ball to go up, the time to
come back down is
(A) less
(B) equal
(C) greater
73. When a bullet is fired from a gun, the bullet and the gun
have equal and opposite momenta. If this is true, then why
is the bullet deadly, but you can hold the gun while it is
fired?
(A) the bullet is much sharper than the gun
(B) the bullet is smaller and can penetrate the body
(C) the bullet has more kinetic energy than the gun
(D) the bullet goes a longer distance and gains speed
74. A rock is dropped from 20 m. What is the final velocity?
a. Use kinematics to solve this problem.
b.
67. A box sliding on a frictionless flat surface runs into a fixed
spring, which compresses a distance x to stop the box. If
the initial speed of the box is doubled, how much would the
spring compress?
(A) ½x
(B) x
(C) 2x
(D) 4x
Use conservation of energy to solve this problem.
75. A pendulum bob reaches a maximum height of 0.6 m above
the lowest point in the swing, what is its fastest speed?
76. How far must a 1 kg ball fall in order to compress a spring
0.1 m? (k = 1000 N/m)
77. A 10-kg box is initially at the top of a 5-m long ramp set at
53o. The box slides down to the bottom of the ramp. The
force of friction is 31 N. Determine the
a. potential energy at the top of the ramp.
Air Resistance Lab
81. Measure how far a cart rolls down a ramp and across a
straight away with and without a sail.
a. Collect the following data.
cart mass, m
ramp length, L
Small Sail + Slow Velocity
Small Sail + Double Velocity
acceleration time t
1.2 s acceleration time t
0.6 s
floor distance d
floor distance d
Double Sail + Slow Velocity Double Sail + Double Velocity
acceleration time t
1.2 s acceleration time t
0.6 s
floor distance d
floor distance d
b.
b.
work done by friction during the slide.
Calculate the following from the data.
Small
Small
Large
Slow
Fast
Slow
Large
Fast
vo
c.
velocity of the box at the bottom of the ramp.
K
78. A spring (k = 500 N/m) is attached to the wall. A 5-kg
block on a horizontal surface ( = 0.25) is pushed against
the spring so that the spring is compressed 0.2 m. The
block is released and propelled across the surface.
a. Determine the potential energy of the spring.
b.
Ff
c.
Engineers estimate the force of air resistance with the
formula, Fair  Av2, where A is sail area and v is
velocity. Discuss whether the data supports this
formula or not.
d.
For a human body in free fall, Fair = 0.22v2. What is the
maximum speed attained by a falling person?
Determine the distance that the block travels.
79. A 15-g bullet penetrates a 1.1-kg block of wood. As a
result, the block slides along a surface ( = 0.85) for 9.5 m.
a. How much work is done by friction?
b.
What is the velocity of the system just after the impact?
c.
What is the velocity of the bullet just before the impact?
82. Andre hits a 0.06-kg tennis ball straight up into the air with
a 300-N force. The ball remains on the racket for 0.25 m.
a. Using dynamics and kinematics, determine
(1) The acceleration.
(2) The initial velocity.
80. An 18-g bullet traveling at 230 m/s buries itself in a 3.6-kg
pendulum hanging on a 2.8-m long string.
a. Determine for the bullet/pendulum just after impact.
(1) velocity
(3) The maximum height reached by the tennis ball.
b.
(2) kinetic energy
b.
How high does the pendulum rise?
Use conservation of energy to determine the
maximum height reached by the tennis ball.
83. How much power is used to lift 100 kg a distance of 2 m in
4 s?
84. A 1,000-kg car maintains a constant speed of 30 m/s against
a combined friction and air resistance force of 550 N.
a. How much power is needed to cruise at 30 m/s?
b.
Calculate the following from the data.
Formula
Calculation
Pendulum
h
b.
How steep an incline can the car climb if the engine
can generate 50,000 W of power?
Ug
K
85. A 1-kg block is pushed down against a spring (k = 500 N/m),
which is compressed 0.1 m. The block is released and
propelled vertically.
a. Determine the potential energy of the spring.
v'
vA
Kinematics
b.
Determine the maximum height reached by the block.
t
vA
86. A 10-kg box is initially at the top of a 5-m long ramp set at
30o. The box slides down to the bottom of the ramp. The
force of friction is 26 N. Determine the
a. potential energy at the top of the ramp.
Velocimeter
vA
c.
b.
work done by friction during the slide.
Calculate the percent differences with the velocimeter.
Formula
Calculation
P
K
c.
velocity of the box at the bottom of the ramp.
87. A 0.050-kg bullet traveling at 1,000 m/s penetrates a 10-kg
block of wood.
a. What is the velocity of the block after impact?
b.
c.
Practice Multiple Choice (No calculator)
Briefly explain why the answer is correct in the space provided.
1
2
3
4
5
6
7
8
9
10 11 12
D
A
A
D
C
D
A
B
A
D
C
A
13 14 15 16 17 18 19 20 21 22 23 24
B
C
D
C
C
D
C
C
B
D
C
B
25 26 27 28 29 30 31 32 33 34 35
B
B
A
C
A
D
A
B
A
D
C
1. Two pucks, where mI = 3mII, are attached by a stretched
spring and are initially held at rest on a frictionless surface.
How far does the block travel along a rough surface
( = 0.25) before stopping?
The pucks are released simultaneously. Which is the
same for both pucks as they move toward each other?
(A) Speed
(B) Velocity
(C) Acceleration
(D) Magnitude of momentum
How high does the block rise if it were suspended
from a long string?
Ballistic Pendulum Lab
88. Measure the velocity of a projectile using pendulum data,
kinematic data and the velocimeter, and comparing the
three values.
a. Collect the following data.
Pendulum
Kinematics
bearing mass, mA
height, dy
pendulum mass, mB
distance, dx
pendulum angle, 
Velocimeter
lowest height, ho
height at , h
2.
A 2,000-kg railroad car rolls to the right at 10 m/s and
collides and stick to a 3,000-kg car that is rolling to the left at
5 m/s. What is their speed after the collision?
(A) 1 m/s
(B) 2.5 m/s (C) 5 m/s
(D) 7 m/s
3.
A 5-kg block with momentum = 30 kg•m/s, sliding east
across a horizontal, frictionless surface, strikes an obstacle.
The obstacle exerts all impulse of 10 N•s to the west on the
block. The speed of the block after the collision is
(A) 4 m/s
(B) 8 m/s
(C) 10 m/s (D) 20 m/s
4.
5.
In the diagram, a block of mass M initially at rest on a
frictionless horizontal surface is struck by a bullet of mass
m moving with horizontal velocity v.
10. What is the change in gravitational potential energy for a
50-kg snowboarder raised a vertical distance of 400 m?
(A) 50 J
(B) 200 J
(C) 20,000 J (D) 200,000 J
What is the velocity of the bullet-block system after the
bullet embeds itself in the block?
(A) (M + v)m/M
(B) (m + v)m/M
(C) (m + M)v/M
(D) mv/(m + M)
11. How high is a 50-N object moved if 250 J of work is done
against the force of gravity?
(A) 2.5 m (B) 10 m
(C) 5 m
(D) 25 m
A disc of mass m is moving to the right with speed v when
it collides and sticks to a second disc of mass 2m. The
second disc was moving to the right with speed v/2.
12. What is the spring potential energy when a spring (k = 80
N/m) is stretched 0.3 m from its equilibrium length?
(A) 3.6 J
(B) 12 J
(C) 7.2 J
(D) 24 J
The speed of the composite body after the collision is
(A) v/3
(B) v/2
(C) 2v/3
(D) 3v/2
13. What is the kinetic energy of a 5-kg block that slides down
an incline at 6 m/s?
(A) 20 J
(B) 90 J
(C) 120 J
(D) 240 J
6.
An object of mass m is moving with speed vo to the right on
a horizontal frictionless surface when it explodes into two
pieces. Subsequently, one piece of mass 2/5m moves with
a speed ½vo to the left. The speed of the other piece of
the object is
(A) vo/2
(B) vo/3
(C) 7vo/5
(D) 2 vo
Questions 14-15 A weight lifter lifts a mass m at constant speed
to a height h in time t.
14. How much work is done by the weight lifter?
(A) mg
(B) mh
(C) mgh
(D) mght
7.
Two objects of mass 0.2 kg and 0.1 kg, respectively, move
parallel to the x-axis. The 0.2 kg object overtakes and
collides with the 0.1 kg object. Immediately after the
collision, the y-component of the velocity of the 0.2 kg
object is 1 m/s upward.
15. What is the average power output of the weight lifter?
(A) mg
(B) mh
(C) mgh
(D) mgh/t
16. The graphs show the position d versus time t of three
objects that move along a straight, level path.
What is the y-component of the velocity of the 0.1 kg object
immediately after the collision?
(A) 2 m/s downward
(B) 0.5 m/s downward
(C) 0 m/s
(D) 0.5 m/s upward
has no change in kinetic energy?
(A) II only (B) III only (C) I and II (D) I and III
8.
Two particles of equal mass mo
moving with equal speeds vo
along paths inclined at 60° to the
x-axis, collide and stick together.
Their velocity after the collision
has magnitude
(A) vo/4
(B) vo/2
(C) vo/2
(D) 3vo/2
17. Which is a scalar quantity that is always positive or zero?
(A) Power
(B) Work
(C) Kinetic energy
(D) Potential Energy
Questions 18-19 A constant force of 900 N pushes a 100 kg
mass up the inclined plane at a uniform speed of 4 m/s.
9.
Student A lifts a 50-N box to a height of 0.4 m in 2.0 s.
Student B lifts a 40-N box to a height of 0.50 m in 1.0 s.
Compared to student A, student B does
(A) the same work but develops more power
(B) the same work but develops less power
(C) more work but develops less power
(D) less work but develops more power
18. The power developed by the 900-N force is
(A) 400 W (B) 800 W (C) 900 W (D) 3600 W
19. The gain in potential energy when the mass goes from the
bottom of the ramp to the top.
(A) 100 J
(B) 500 J
(C) 1000 J (D) 2000 J
20. What is the maximum height that a 0.1-kg stone rises if 40
J of work is used to shoot it straight up in the air?
(A) 0.4 m (B) 4 m
(C) 40 m
(D) 400 m
21. A 40-N block is released from rest on an incline 8 m above
the horizontal.
26. What does the slope of the graph represent?
(A) mass of the object
(B) gravitational force on the object
(C) kinetic energy of the object
(D) potential energy of the object
27. If an object with greater mass was graphed instead of the
object graphed above, how would the slope of the graph
differ from the above graph?
(A) more positive
(B) less positive
(C) equal but negative
(D) be the same
28. Which is the graph of the spring potential energy of a
spring versus elongation from equilibrium?
(A)
(B)
(C)
(D)
What is the kinetic energy of the block at the bottom of the
incline if 50 J of energy is lost due to friction?
(A) 50 J
(B) 270 J
(C) 320 J
(D) 3100 J
22. A 50-kg diver falls freely from a diving platform that is 3 m
above the surface of the water. What is her kinetic energy
at 1 m above the water?
(A) 0
(B) 500 J
(C) 750 J
(D) 1000 J
23. A 1000 W electric motor lifts a 100 kg safe at constant
velocity. The vertical distance through which the motor
can raise the safe in 10 s is most nearly
(A) 1 m
(B) 3 m
(C) 10 m
(D) 32 m
29. Which is the graph of the gravitational potential energy of
an object versus height? (Assume height << earth's radius)
(A)
(B)
(C)
(D)
30. An object falls freely near earth's surface. Which graph
best represents the relationship between the object's
kinetic energy and its time of fall?
(A)
(B)
(C)
(D)
Questions 24-27 The vertical height versus gravitational potential
energy for an object near earth's surface is graphed below.
Ug (J)
80
60
40
20
0
h (m)
0.5
1.5
2.5
3.5
24. What is Ug when the object is 2.25 m above the surface?
(A) 50 J
(B) 45 J
(C) 60 J
(D) 55 J
25. What is the mass of the object?
(A) 1.5 kg (B) 2.0 kg (C) 2.5 kg
(D) 3.0 kg
31. A system consists of two masses m1 and m2
(m1 < m2). The objects are connected by a string,
hung over a pulley and then released. When the
object of mass m2 has descended a distance h,
the change in potential energy of the system is
(A) (m1 – m2)gh
(B) m2gh
(C) (m1 + m2)gh
(D) ½(m1 + m2)gh
32. From the top of a 70-m-high building, a 1-kg ball is thrown
directly downward with an initial speed of 10 m/s. If the
ball reaches the ground with a speed of 30 m/s, the energy
lost to friction is most nearly
(A) 50 J
(B) 250 J
(C) 300 J
(D) 450 J

33. A block of mass M is released from rest at the top of an
inclined plane of height h. If the plane is frictionless, what
is the speed of the sphere at the bottom of the incline?
(A) (2gh)½ (B) 2Mgh
(C) 2MghR2 (D) 5gh
Determine the work done by this force to move the object
a. from x = 0 to x = 10 m

b.
34. Block of mass m slides on a horizontal frictionless table
with an initial speed vo. It then compresses a spring of
force constant k and is brought to rest. How much is the
spring compressed from its natural length?
(A) vo2/2g (B) mgvo/k (C) mvo/k
(D) vo(m/k)½
35. A small mass is released from rest
U
at a very great distance from a larger
stationary mass. Which graph
0
represents the gravitational potential
energy U of the system of the two
masses as a function of r?
1.
4.
Students are to calculate the spring constant k of a spring
that initially rests on a table. When the spring is
compressed a distance x from its uncompressed length Lo
and then released, the top rises to a maximum height h
above the point of maximum compression. The students
repeat the experiment, measuring h with various masses m
taped to the top of the spring.
(A)
r
(B)
(C)
Practice Free Response
A 0.62-kg block is attached to the spring (k = 180 N/m).
When the system is compressed 5.0 cm and released, it
slides a total of 7.3 cm before turning back.
What is the coefficient of friction?
2.
from x = 0 to x = 15 m
a.
Derive an expression for the height h in terms of m, x,
k, and fundamental constants.
With the spring compressed a distance x = 0.020 m in each
trial, the students obtained the data for different values of m.
Block with mass M slides down and strikes a smaller block
with mass m, where M = 3m. The blocks stick and fall to
the floor.
m (kg)
h (m)
b.
c.
0.020
0.49
0.030
0.34
0.040
0.28
0.050
0.19
0.060
0.18
(1) What quantities should be graphed so that the
slope of a best-fit straight line can be used to
calculate the spring constant k?
(2) Fill in one or both of the blank columns in the
table with calculated values of your quantities.
On the axes below, plot your data and draw a best-fit
straight line. Label the axes and indicate the scale.
How far horizontally from the table's edge do the blocks
land?
3.
The force acting on an object along the x axis varies as
shown.
d.
Using your best-fit line, calculate the spring constant.
e.
The height h that the spring reaches is difficult to
measure. How would you determine this value?
Fx (N)
400
200
0
-200
1
3
5
7
9
11
13
15
x (m)