Conservation Laws
Additional Practice Problems
AP Physics 1
Kuffer
Unit 4 Outline: Work, Energy, Momentum, and Impulse
Due
Date
Class
Number
57
60
63
64
67
68
70
73
Topic
Work done by a constant force
Work done by a variable force
Mechanical energy and
work energy theorem
Conservative and nonconservative
forces
Conservation of mechanical energy
Book
Section
(s)
6.1
6.9,
10.1
6.2-6.3
6.4
6.5-6.6
Power
6.7
Impulse and Momentum
Conservation of Momentum
7.1
7.2
Collisions and Coefficient of
Restitution
Center of Mass
7.3-7.4
7.5
Assignment
Video(s)
FOC: 1-2
P: 1-4, 6-12
Ch 6: P: 71, 74, 75
Ch 10: FOC: 2; P: 1-3, 25-26
FOC: 8-9
P: 13-15, 17-18, 20, 22, 24, 28
FOC: 11, 13-14, 21-22
P: 29-31, 33-34, 36-40, 42,
44-45, 47-53, 57, 59-61
4.1-4.3
4.4-4.5
4.6-4.9
4.104.12
4.134.16
4.17
FOC: 25
P: 62-64, 67, 69
FOC: 1-2, 6-7, 10
P: 1-2, 5-7, 10-12, 15-17,
19-22, 24-27
FOC: 13, 15
P: 28, 30, 32-39, 42, 44-48
FOC: 17
P: 48-50, 62, 64
4.18
4.19
4.204.23
4.24
1
WORK & POWER
2
3
4
ENERGY
5
6
7
8
9
10
Work and Power
1. Brutus, a champion weightlifter, raises 260 kg of weights a distance of 2.25 m.
a) How much work is done by Brutus?
b) How much work is done by Brutus holding the
weights above his head?
c) How much work is done by Brutus lowering
them back to the ground?
d) Does Brutus do work if he lets go of the weights
and they fall to the ground?
e) If Brutus completes the lift in 1.9 s, how much power is developed?
2. A horizontal force of 805 N is needed to drag a crate across a horizontal floor with a
constant speed. You drag the crate using a rope held at an angle of 32.
a) What force do you exert on the rope?
b) How much work do you do on the crate when moving 22 m?
c) If you complete the job in 8.0 s, what power is developed?
3. An airplane passenger carries a 223 N suitcase up the stairs, a displacement of 5.3 m
horizontally and 5.6 m vertically.
a) How much work does the passenger do?
b) The same passenger carries the same suitcase back down the same stairs. How
much work does the passenger do now?
11
WORK, POWER, KINETIC ENERGY & POTENTIAL ENERGY
Description
+ or Work?
Change PE or
KE or Both?
Megan drops the ball and hits an awesome forehand. The racket is moving
horizontally as the strings apply a horizontal force while in contact with the
ball.
A baseball player hits the ball into the outfield bleachers. During the contact
time between ball and bat, the bat is moving at a 10 degree angle to the
horizontal.
Rusty Nales pounds a nail into a block of wood. The hammer head is moving
horizontally when it applies force to the nail.
The frictional force between highway and tires pushes backwards on the tires
of a skidding car.
A diver experiences a horizontal reaction force exerted by the blocks upon her
feet at start of the race.
A weightlifter applies a force to lift a barbell above his head at constant
speed.
12
Description of Motion
KE to PE or PE to KE?
Explain.
1.
A ball falls from a height of 2
meters in the absence of air
resistance.
2.
A skier glides from location A to
location B across the friction free
ice.
3.
A baseball is traveling upward
towards a man in the bleachers.
4.
A bungee chord begins to exert an
upward force upon a falling bungee
jumper.
5.
The spring of a dart gun exerts a
force on a dart as it is launched from
an initial rest position.
13
Work, Energy, and Power Study Guide
Name:____________
1. Work is done only if the object pushed is ___________ .
2. Potential energy depends upon what three variables?
a. ________________
b. ________________
c. ________________
3. What variables do momentum and kinetic energy have in common?
a. ________________
b. ________________
4. Energy is the ability of an object to make a ____________ in itself or the
environment.
5. List several different types of energy.
a. ________________
b. ________________
c. ________________
d. ________________
6. ____________ is the process of changing the energy of the system.
7. Work is only done if the object is displaced by the__________, in the same
direction as the _________! (same word for both blanks)
8. ___________ is the rate at which work is done.
9. Draw before and after pictures of a collision containing two motion carts of the
same mass. The carts stick together after the collision.
10. Represent the above picture with an equation.
11. The above equation is referred to as the Law of ___________of
_______________.
12. Power is measured in ___________.
13. Energy is measured in ___________.
14. Work done on a system results in ________ energy.
15. Stored energy is referred to as ________ energy.
16. The work-energy theorem states that ________ is equal to _________.
17. Sometimes we say that the energy was lost due to several different variables. Is
the energy truly lost? ____ Where does it go?
____________________________________________________
14
18. When an object is dropped, it immediately begins to transfer ___________ energy
to ___________ energy.
19. A skier starts from rest at the top of a 45-m hill, skies down at 30 degrees into a
valley, and continues up a 40-m hill. Both hill heights are measured from the
valley floor. Neglect friction.
a. How fast is the skier moving at the bottom of the first hill?
b. What is the skier’s speed at the top of the next hill?
c. Do your answers depend on the angle of the hill? _____ Why, Why not?
________________________________________
_______________________________________________________________
_____________________________________
20. A 4200-N piano is slid up a 3.5-m plank at a constant speed. The plank makes an
angle of 30 degrees with the horizontal.
a. What is the total height the piano is displaced?
b. Calculate the work done by the person sliding it up the incline.
c. If the worker does the job in 60 seconds, what power do they generate?
21. A 90-kg rock climber first climbs 45-m up to the top of a quarry, then descends
85-m from the top to the bottom of the quarry. If the initial height is the reference
level, find the potential energy of the climber at the top and the bottom. Draw bar
graphs for both situations.
15
Additional Problems
Part I - Mechanical Energy
W  F d
PE  mgh
KE 
1
mv 2
2
W  KE
1. Identical twins Pat and Chris are painting a house. Pat is standing on the scaffolding
5 meters above the ground. Chris is standing on the scaffolding 5 meters above Pat.
Who has more potential energy? Explain.
2. Jared and Clay are climbing the stairs. Jared gets tired and stops halfway to the
fourth floor. Clay makes it to the fourth floor without a problem. If Jared is twice as
heavy as Clay, who has more potential energy? Explain.
3. A person weighing 630 N climbs up a ladder to a height of 5 meters. How much
work does the person do? Determine the increase in the potential energy of the
person from the ground to this height. Where does the energy come from to cause
this increase in PE?
4. Calculate the kinetic energy of a 750 kg car moving at 13.9 m/s. What is the kinetic
energy of the car if the speed is doubled? How much work must be done to double
the speed?
5. A rifle can shoot a 4.2 g bullet at a speed of 965 m/s. Find the kinetic energy of the
bullet. What work is done on the bullet if it starts from rest? If the work is done over
a distance of 75 cm, determine the average force acting on the bullet.
16
Part II – Problem Solving with Conservation of Energy
Eo  E f
PE  mgh
KE 
1
mv 2
2
1. A large chunk of ice with mass 15 kg falls from a roof 8 meters above the ground.
Find the kinetic energy of the ice when it reaches the ground. What is the speed of
the ice when it reaches the ground?
2. A bike rider approaches a hill with a speed of 8.5 m/s. The total
mass of the bike and the rider is 85 kg. Find the kinetic energy
of the bike and rider. If the rider coasts up the hill, calculate the
height at which the bike will come to a stop. (Assume there is
no friction.) How would your answer vary if the mass of the
bike and rider were doubled?
3. A 2 kg rocket is launched straight up into the air with a speed that allows it to reach a
height of 100 meters, even though air resistance performs 800 J of work on the
rocket. Determine the launch speed of the rocket. How high would the rocket travel
if air resistance is ignored?
17
18
4. Calculate the potential energy, kinetic energy, mechanical energy, velocity, and height of the skater at the various locations.
Part III – Problem Solving with Conservation of Energy
Eo  E f
o   f
PE  mgh
KE 

1
mv 2
2

  mv
1. A 20 kg rock is on the edge of a 100 meter cliff. Calculate the potential energy of the
rock. If the rock falls off the cliff, what is its kinetic energy just before striking the
ground? What speed does the rock have as it strikes the ground?
2. A physics book is dropped 4.5 meters. What speed does the book have just before it
hits the ground?
3. From what height would a compact car have to be dropped to have the same kinetic
energy that is has when being driven at 100 km/hr?
4. A 70 kg high jumper leaves the ground with a speed of 6 m/s. How high can he
jump?
5. Just before striking the ground, a 900 kg Smart Bomb has 88.2 MJ of kinetic energy.
If air resistance is ignored, determine the height from which the Smart Bomb was
dropped. Determine the drop height if air resistance performs 8.82 MJ of work
against the bomb as it falls towards its target.
6. A 74 kg student, starting from rest, slides down an 11.8 meter high water slide. On
the way down, friction does 5600 J of work on him. How fast is he going at the
bottom of the slide?
7. Block A with a mass of 12 kg moving at 2.4 m/s makes a perfectly elastic head-on
collision with block B, mass 36 kg, at rest. Find the velocities of the two blocks after
the collision. Assume all motion is in one dimension.
19
20
8. Calculate the potential energy, kinetic energy, mechanical energy, velocity, and height of the ball at the various locations.
Part IV - Chapter Review
W  F d
PE  mgh
Eo  E f
W  KE
KE 
1
mv 2
2
1. Jamie lifts her toys into her tree house using a homemade elevator. The elevator has
a mass of 2.5 kg and the tree house is 8 meters above the ground. How much work
does Jamie do when lifting 5 kg of toys into the house? Determine the power used to
lift the toys in 5 sec.
2. Mike pulls a sled across level snow with a force of 225 N using a rope that is angled
at 35. Determine the work done if he pulls the sled 65.3 meters.
3. A 2 kg textbook is lifted from the floor to a shelf 2.1 meters above the floor.
Determine the book’s potential energy relative to the floor. What is the book’s
potential energy relative to the head of a 1.65 meter tall person?
4. A shot-putter heaves a 7.26 kg shot with a velocity of 7.5 m/s. Determine the kinetic
energy of the shot. How much work was done on the shot to give it its kinetic
energy?
5. Calculate the kinetic energy of a 750 kg compact car moving at 100 km/hr. How
much work must be done to slow the car down to 50 km/hr?
6. Determine the mechanical energy of a 450 kg roller coaster moving at 30 m/s at the
bottom of the first dip which is 15 meters above the ground.
7. Julie has a mass of 49 kg. What is her potential energy when standing on the 6 meter
diving board? (She is 6 meters above the water.) Julie jumps off the diving board.
What is her kinetic energy right before she hits the water? How fast does Julie hit the
water?
8. An unfortunate skydiver’s parachute fails to open. If the diver hits the ground going
300 m/s, determine the height from which the ill-fated jump was make.
21
22
9. Calculate the potential, kinetic, and mechanical energies, velocity, work, and power of the ball at the various locations.
Where did the term horsepower originate?
The term horsepower came from Scottish inventor James Watt. The
value for a unit of horsepower was determined after Watt performed
several experiments on horses pulling coal. He originally determined
that the average horse was able to pull 22,000 foot-pounds every one
minute. In other words, a horsepower was defined as the amount of
power exerted to move 22,000 pounds of coal by one foot in one minute.
Watt was not happy with his figure because he felt it was too low; he
thought the average horse was more powerful than his original
calculations and experiment indicated, so after extensive study of horses,
he increased the value of a horsepower to 33,000 foot-pounds per
minute. (1 HP = 746 W)
What countries are the largest consumers of energy?
23
What are the average yearly costs of some general home appliances?
Home climate control and appliances account for approximately
one third of the energy consumption in the United States. The
average cost for energy is approximately $0.135 per KwH
(kilowatt-hour) in PA… but varies throughout the country. The
following is a listing of home appliances, their typical usage, and
the cost for one full year.
Appliance
Television
Energy (KwK) Annual Cost @ $0.12 / Kw
1000
$120
1000
150
1000
1200
2000
5000
1500
$120
$18
$120
$152
$240
$600
$180
(8 hours per day)
Stove with Oven
Washer
Dryer
Refrigerator
Frost-free Refrigerator
Hot-water Heater
Window Air-conditioner
(if used year ‘round)
24
14.
 A father pulls his young daughter on a sled with a constant velocity on a level surface through a
distance of 10 m, as illustrated in Fig. 5.25a. If the total mass of the sled and the girl is 35 kg and
the coefficient of kinetic friction between the sled runners and the snow is 0.20, how much work does
the father do? 6.1  102 J
25
26
65.
 A 1.50-kg mass is placed on the end of a spring that has a spring constant of 175 N m. The
mass-spring system rests on a frictionless incline that is at an angle of 30° from the horizontal (Fig.
5.28). The system is eased into its equilibrium position, where it stays. (a) Determine the change in
elastic potential energy of the system. (b) Determine the system’s change in gravitational potential
energy. (a) 0.154 J (b) 0.309 J
65.
(a) The component of the weight, mg, of the object along the incline (parallel to
the spring) is equal to
mg sin . (See Exercise 4-69.) This is the force that stretches the spring.
mg sin
F
x= s =
k
k

Us =
k xo2 =
1
2
k x2 
1
2
(1.50 kg)(9.80 m/s2) sin 30
=
= 0.0420 m.
175 N/m
1
2
(175 N/m)(0.0420 m)2  0 = 0.154 J .
(b) The vertical distance the mass moves down is equal to x sin  = (0.0420 m)
sin 30 = 0.0210 m.
Ug = mgy = (1.50 kg)(9.80 m/s2)(0.0210 m  0) = 0.309 J .
27
86.
 A hiker plans to swing on a rope across a ravine in the mountains, as illustrated inFig. 5.33,
and to drop when she is just above the far edge. (a) At what horizontal speed should she be moving
when she starts to swing? (b) Below what speed would she be in danger of falling into the ravine?
Explain. (a) at least 2.9 m s (b) starting with vo  2.9 m s
Sinθ=1.8/4.0
Θ = 26.7°
4.0-3.57 = h
H = .43 m
Θ
86 .
(a)  = sin1 
18
. 
 = 26.7.
 4.0 
4.0
3.57
So
h = 0.427 m. 
mgh = 12 m v2o … (“m’s” cancel)
So
vo =
2gy =
2(9.80 m/s2)(0.427 m) = V = 2.9 m/s.
1.8
Therefore the speed has to be at least 2.9 m/s .
(b) If starting with vo < 2.9 m/s , she would be in danger of falling into the ravine
because of insufficient energy.
28
81.
 A roller coaster travels on a frictionless track as shown in Fig. 5.31. (a) If the speed of the
roller coaster at point A is 5.0 m s, what is its speed at point B? (b) Will it reach point C? (c) What
minimum speed at point A is required for the roller coaster to reach point C? (a) 11 m s (b) no (c)
7.7 m s
81.
Choose the lowest point on the course (point B) as the reference for height (yo =
0).
(a) 12 m vB2 + UB = 12 m vA2 + UA,
so
vB =

1
2
m vB2 + mg(0) = 12 m(5.0 m/s)2 + mg(5.0 m),
(5.0 m/s)2 + 2(9.80 m/s2)(5.0 m) = 11 m/s .
(b) EA = EB = 12 m (11 m/s)2 = 60.5m.
EC = mg(8.0 m) = m(9.80 m/s2)(8.0 m) =
78.4m > EA.
So no , it will not reach point C.
(c) 12 m vA2 + mg(5.0 m) = 12 m(0)2 + mg(8.0 m),

vA = 7.7 m/s .
Conservative and Nonconservative Forces and The Conservation of ME
29
**Even though there is a nonconservative force, tension… it does no work… Note: the
tension force is perpendicular to the path (aka displacement) of the boy at every instant…
therefore no work is done by the nonconservative force!!
30
IMPULSE & MOMENTUM
1. Explain why it is difficult for a firefighter to hold a hose which ejects large
amounts of high-speed water.
________________________________________________
________________________________________________
________________________________________________
________________________________________________
2. A large truck and a Volkswagen have a head-on collision.
a. Which vehicle experiences the greatest force of impact?
Truck or VW
b. Which vehicle experiences the greatest impulse?
Truck or VW
c. Which vehicle experiences the greatest momentum change?
Truck or VW
d. Which vehicle experiences the greatest acceleration?
Truck or VW
3. Miles Tugo and Ben Travlun are riding in a bus at highway speed on a nice
summer day when an unlucky bug splatters onto the windshield. Miles and Ben begin
discussing the physics of the situation. Miles suggests that the momentum change of
the bug is much greater than that of the bus. After all, argues Miles, there was no
noticeable change in the speed of the bus compared to the obvious change in the
speed of the bug. Ben disagrees entirely, arguing that that both bug and bus
encounter the same force, momentum change, and impulse. Who do you agree with?
Support your answer.
________________________________________________
________________________________________________
________________________________________________
________________________________________________
________________________________________________
________________________________________________
4. If a ball is projected upward from the ground with ten units of momentum, what is
the momentum of recoil of the Earth? ____________ Do we feel this? __________
Explain.
________________________________________________
________________________________________________
________________________________________________
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5. Would you care to fire a rifle that has a bullet ten times as massive as the rifle?
YES or NO
Explain.____________________________________________
________________________________________________
________________________________________________
8. A baseball player holds a bat loosely and bunts a ball. Express your understanding
of momentum conservation by filling in the tables below.
a. _____________
b. _____________
c. _____________
9. A Tomahawk cruise missile is launched from the barrel of a mobile missile
launcher. Neglect friction. Express your understanding of momentum conservation by
filling in the tables below.
a. _____________
b. _____________
c. _____________
32
Check Your Understanding I
AP Physics 1 - Mr. Kuffer
Express your understanding of the concept and mathematics of momentum by answering
the following questions.
1. Determine the momentum of a ...
a. 60-kg halfback moving eastward at 9 m/s.
b. 1000-kg car moving northward at 20 m/s.
c. 40-kg freshman moving southward at 2 m/s.
2. A car possesses 20 000 units of momentum. What would be the car's new momentum
if ...
a. its velocity were doubled.
b. its velocity were tripled.
c. its mass were doubled (by adding more passengers and a greater load)
33
3. A halfback (m = 60 kg), a tight end (m = 90 kg), and a lineman (m = 120 kg) are
running down the football field. Consider their ticker tape patterns below.
Compare the velocities of these three players. How many times greater is the velocity of
the halfback than the velocity of the lineman?
How many times greater is the velocity of the tight end than the velocity of the lineman?
Which player has the greatest momentum?
Lineman or Tight End or Halfback
Explain.
________________________________________________
________________________________________________
________________________________________________
4. For the next three vector diagrams, write “A” or “B” next to each question.
Vector Diagram
Greatest velocity change?
Greatest acceleration?
Greatest momentum change?
Greatest Impulse?
________________________________________________________________________
34
Velocity-Time Graph
Greatest velocity change?
Greatest acceleration?
Greatest momentum change?
Greatest Impulse?
________________________________________________________________________
Ticker Tape Diagram
Greatest velocity change?
Greatest acceleration?
Greatest momentum change?
Greatest Impulse?
35
Check Your Understanding Part II
AP Physics 1 - Mr. Kuffer
Express your understanding of the impulse-momentum change theorem by answering the
following questions.
1. A 0.50-kg cart (#1) is pulled with a 1.0-N force for 1 second; another 0.50 kg cart (#2)
is pulled with a 2.0 N-force for 0.50 seconds. Which cart (#1 or #2) has the greatest
acceleration? Explain.
Which cart (#1 or #2) has the greatest impulse? Explain.
Which cart (#1 or #2) has the greatest change in momentum? Explain.
2. In a phun physics demo, two identical balloons (A and B) are propelled across the
room on horizontal guide wires. The motion diagrams (depicting the relative position of
the balloons at time intervals of 0.05 seconds) for these two balloons are shown below.
Which balloon (A or B) has the greatest acceleration? Explain.
Which balloon (A or B) has the greatest final velocity? Explain.
Which balloon (A or B) has the greatest momentum change? Explain.
Which balloon (A or B) experiences the greatest impulse? Explain.
36
3. Two cars of equal mass are traveling down Lake Avenue with equal velocities. They
both come to a stop over different lengths of time. The ticker tape patterns for each car
are shown on the diagram below.
At what approximate location on the diagram (in terms of dots) does each car begin to
experience the impulse.
Which car (A or B) experiences the greatest acceleration? Explain.
Which car (A or B) experiences the greatest change in momentum? Explain.
Which car (A or B) experiences the greatest impulse? Explain.
4. The diagram to the right depicts the before- and after-collision speeds of a car which
undergoes a head-on-collision with a wall. In Case A, the car bounces off the wall. In
Case B, the car "sticks" to the wall.
In which case (A or B) is the change in velocity the greatest? Explain.
In which case (A or B) is the change in momentum the greatest? Explain.
In which case (A or B) is the impulse the greatest? Explain.
In which case (A or B) is the force which acts upon the car the greatest (assume contact
times are the same in both cases)? Explain.
37
5. Rhonda, who has a mass of 60.0 kg, is riding at 25.0 m/s in her sports car when she
must suddenly slam on the brakes to avoid hitting a dog crossing the road. She strikes the
air bag, which brings her body to a stop in 0.400 s. What average force does the seat belt
exert on her?
If Rhonda had not been wearing her seat belt and not had an air bag, then the windshield
would have stopped her head in 0.001 s. What average force would the windshield have
exerted on her?
6. A hockey player applies an average force of 80.0 N to a 0.25 kg hockey puck for a
time of 0.10 seconds. Determine the impulse experienced by the hockey puck.
7. If a 5-kg object experiences a 10-N force for a duration of 0.1-second, then what is the
momentum change of the object?
38
Check Your Understanding Part III
AP Physics 1 - Mr. Kuffer
Express your understanding of Newton's third law by answering the following questions.
1. While driving down the road, an unfortunate bug strikes the windshield of a bus. Quite
obviously, a case of Newton's third law of motion. The bug hit the bus and the bus hit the
bug. Which of the two forces is greater: the force on the bug or the force on the bus?
2. Rockets are unable to accelerate in space because ...
a. there is no air in space for the rockets to push off of.
b. there is no gravity is in space.
c. there is no air resistance in space.
d. ... nonsense! Rockets do accelerate in space.
3. A gun recoils when it is fired. The recoil is the result of action-reaction force pairs. As
the gases from the gunpowder explosion expand, the gun pushes
the bullet forwards and the bullet pushes the gun backwards. The
acceleration of the recoiling gun is ...
a. greater than the acceleration of the bullet.
b. smaller than the acceleration of the bullet.
c. the same size as the acceleration of the bullet.
39
4. Would it be a good idea to jump from a rowboat to a dock that seems within jumping
distance? Explain.
5. If we throw a ball horizontally while standing on roller skates, we roll backward with a
momentum that matches that of the ball. Will we roll backward if we go through the
motion of throwing the ball without letting go of it? Explain.
6. Suppose there are three astronauts outside a spaceship and two of them decide to play
catch with the other woman. All three astronauts weigh the same on Earth and are equally
strong. The first astronaut throws the second astronaut towards the third astronaut and the
game begins. Describe the motion of these women as the game proceeds. Assume each
toss results from the same-sized "push." How long will the game last?
40
41
42
Momentum in Two Dimensions
1. Two cars are traveling toward each other in a demolition derby, as shown below.
After they collide they will become entangled together, but no parts will be lost.
A
What is the momentum of Car A before the collision?
mass = 650 kg
velocity = 14 m/s South
What is the momentum of Car B before the collision?
B
mass = 800 kg
velocity = 18 m/s East
What is the mass of the tangled mess of cars after the collision?
What is the momentum of the tangled mess of cars after the collision?
2. This 1.0 kg, lit firecracker explodes while falling at 20. m/s. It splits into two
equal halves.
What is the momentum of the firecracker before exploding?
Piece A is moving with velocity components
of 15 m/s down and 10 m/s left.
What is the resultant velocity of Piece A?
A
B
What are the velocity components of Piece B?
What is the resultant velocity of Piece B?
Elastic vs Inelastic Collisions
43