The Physics
Hypertextbook
Opus in profectus

# Conservation of Energy

## Problems

### practice

1. The diagram below shows a 10,000 kg bus traveling on a straight road which rises and falls. The horizontal dimension has been foreshortened. The speed of the bus at point A is 26.82 m/s (60 mph). The engine has been disengaged and the bus is coasting. Friction and air resistance are assumed negligible. The numbers on the left show the altitude above sea level in meters. The letters A–F correspond to points on the road at these altitudes.

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1. Find the speed of the bus at point B.
2. An extortionist has planted a bomb on the bus. If the speed of the bus falls below 22.35 m/s (50 mph) the bomb will explode. Will the speed of the bus fall below this value and explode? If you feel the bus will explode, identify the interval in which this occurs.
3. Derive an equation to determine the speed of the bus at any altitude.
2. Two 64 kg stick figures are performing an extreme blob jump as shown in the diagram below. (Warning: These are professional stunt stick figures. Don't try this at home.) One stick figure stands atop a 7.0 m high platform with a 256 kg boulder. A second stick figure stands a partially inflated air bag known as a blob (or water trampoline). The first stick figure rolls the boulder off the edge of the platform. It falls onto the blob, catapulting the second stick figure into the air. What is the maximum height to which the second stick figure can rise? Assume that stick figures, boulders, and blobs obey the law of conservation of energy.
3. The illustration below shows a vertical loop segment of a roller coaster. The path of the rails is highlighted in yellow.

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Segments of roller coaster track are rarely circular. The transition between a straight segment and a circular segment, or two circular segments of different radii, would subject the rider to abrupt changes in acceleration, called jerk, that would be uncomfortable, especially at high speeds. Thrill rides should be thrilling, not jarring, jerking, or jostling. Curves with a gradually changing radius of curvature are more common.

The illustration below shows the same vertical loop with circles added to represent the instantaneous curvature at three locations. (For those of you who like technical language, these are called osculating circles.) The illustration also includes the radius of curvature and height above the lowest point on the track for these locations.

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Summary of data given for this problem
segment h (m) r (m)
a. top of loop 25.8 07.1
b. bottom of approach 00.0 23.2
c. vertical descending side 14.0 13.4

Assume minimal energy losses due to air resistance, rolling resistance, or other forms of friction and answer the following questions.

1. Determine the speed of the coaster at the top of the loop if the normal force of the rails on the wheels is half the weight of the coaster (that is, if the frame of reference acceleration is ½g).
2. Determine the speed of the coaster at its lowest point before it entered the loop. How does the normal force on the wheels compare to the weight of the coaster now (that is, what is the new frame of reference acceleration)?
3. Determine the speed of the coaster on the side of the loop when it is instantaneously moving straight down. How does the normal force on the wheels compare to the weight of the coaster at this location (that is, what is the frame of reference acceleration here)?
4. Write something completely different.

### conceptual

1. Four identical balls are thrown from the top of a cliff, each with the same speed. The first is thrown straight up, the second is thrown at 30° above the horizontal, the third at 30° below he horizontal, and the fourth straight down. How do the speeds and kinetic energies of the balls compare as they strike the ground…
1. when air resistance is negligible?
2. when air resistance is significant?
2. A ball is dropped from the top of a tall building and reaches terminal velocity as it falls. (At terminal velocity drag equals weight and a falling object stops accelerating.) Will the potential energy of the ball upon release equal the kinetic energy it has when striking the ground? Explain your reasoning.

### numerical

1. A 55 kg human cannonball is shot out the mouth of a 4.5 m cannon with a speed of 18 m/s at an angle of 60°. (Friction and air resistance are negligible in this problem. You may not use Newton's laws or the equations of motion to solve these problems. Think conservation of energy.) Determine the following quantities for the human cannonball she exits the mouth of the cannon…
1. the horizontal and vertical components of her velocity
2. her kinetic energy
Determine the following quantities for the human cannonball at the top of her trajectory.
1. the horizontal and vertical components of her velocity
2. her kinetic energy
3. her potential energy relative to the mouth of the cannon
4. her height above the mouth of the cannon
2. Watch the video below before beginning this problem. The height of the building Spider-Man (a.k.a. Peter Parker, a.k.a. Tobey McGuire) starts off on is 6 stories, or 18 meters high (assuming one story is 3 meters). The height of the building he wants to swing to is 1 story, or 3 meters high. The crane onto which he shoots his web is 7 stories, or 21 meters high. Tobey McGuire is 1.75 m tall and approximately 72 kg in mass. (Do not use the equations of motion to solve any part of this problem.)

Determine…
1. Spider-Man's speed when his feet touch the roof the second building
2. the maximum tension in Spider-Man's web during this video clip
3. the approximate kinetic energy dissipated when Spider-Man struck the wall
3. A laboratory cart (m1 = 500 g) is pulled horizontally across a level track by a lead weight (m2 = 25 g) suspended vertically off the end of a pulley as shown in the diagram below. (Assume the string and pulley contribute negligible mass to the system and that friction is kept low enough to be ignored.) If the lead weight falls 85 cm, determine…
1. the final speed of the system
2. the acceleration of the system
3. the tension in the string
4. A laboratory cart (m1 = 500 g) rests on an inclined track (θ = 9°). It is connected to a lead weight (m2 = 100 g) suspended vertically off the end of a pulley as shown in the diagram below. (Assume the string and pulley contribute negligible mass to the system and that friction is kept low enough to be ignored.) If the lead weight falls 85 cm, determine…
1. the final speed of the system
2. the acceleration of the system
3. the tension in the string
5. A 45 kg box is pushed up a 21 m ramp at a uniform speed. The top of the ramp is 3.0 m higher than the bottom.
1. What is the potential energy of the box at the top of the ramp relative to the bottom of the ramp?
2. What work was done pushing the box up the ramp if friction were negligible?
3. What work was done pushing the box up the ramp if the force of friction between the box and the ramp was 100. N?
6. A 1200 kg car driving downhill goes from an altitude of 70 m to 40 m above sea level and accelerates from 11 m/s to 23 m/s.
1. How much potential energy did the car lose?
2. How much kinetic energy did it gain?
3. How much energy is unaccounted for?
4. Where did this energy go?
7. An 82 kg skydiver jumps from a height of 95 m and strikes the ground with a speed of 6.0 m/s.
1. Calculate the work done by air resistance on the skydiver.
2. What was the average air resistance on the skydiver during this jump?
3. How does the magnitude of the average air resistance compare to the weight of the skydiver?
8. Top pole vaulters have a mass of about 80 kg and can clear a bar 6.0 m above the ground. Top sprinters also have a mass of about 80 kg and can cover 100 m in 10 s. Given these numbers, show that world record pole vaults would not be possible without the pole contributing some elastic potential energy.
9. A 2.0 kg rock initially at rest loses 400 J of potential energy as it falls freely to the ground.
1. Calculate the kinetic energy that the rock gains while falling.
2. What is the rock's speed just before it strikes the ground?
10. A batter hits a pop fly in baseball. It travels straight up at 26.6 m/s.
1. Using the appropriate equation of motion, determine…
1. the maximum height of the ball above the bat.
2. the time it takes for the ball to reach its highest point.
2. Given that a baseball has a mass of 145 g, determine…
1. the kinetic energy of the ball when it left the bat.
2. the potential energy of the ball at its highest point.

### algebraic

1. A mass m slides down a loop-the-loop track with the dimensions shown in the diagram below. If the mass is released from rest at the point labeled "a" and the track is effectively frictionless, determine the speed of the mass at each of the other five lettered points in terms of m, r, and g. 2. A lab cart of mass m is pulled by a block of mass m in a lab experiment like the one shown in the diagram to the right. Initially, the cart is a distance 2h from the edge of the table and the block is a distance h above the floor. The system starts from rest and is released. Do not use Newton's Second law to solve any part of this problem. State your answers in terms of m, g, and h. Assume that friction is negligible.
1. Determine the speed of the cart when the block lands on the floor
2. Determine the speed of the cart when the cart reaches the edge of the table.

### statistical

1. A crude physical model of the pole vault assumes that all the vaulter's kinetic energy on approach is converted to gravitational potential energy at the top of the vault. As we all know, real world situations are never this simple. If we compare the kinetic energy of an Olympic sprinter to the gravitational potential energy of an Olympic pole vaulter, we find that the two numbers are not equal. In the earlier years of the modern Olympics, the potential energy of a vaulter was always less than the kinetic energy of a sprinter. (No surprise there. Lost energy is inevitable.) Somewhere in the middle of the 20th century, however, the situation reversed. The potential energy of world class pole vaulters now routinely exceeds the kinetic energy of world class sprinters. It would appear that vaulters have discovered a way to "violate" the law of conservation of energy.
1. Using one of the data sets provided below, produce a graph that can be used to identify the year in which the maximum gravitational potential energy of Olympic pole vaulters exceeded the average kinetic energy of Olympic sprinters. (Choose an appropriate mass for an athlete and be sure to identify the year of the transition.)
2. What changed about the sport that enabled pole vaulters to "violate" the law of conservation of energy? (Was it the shoes? Energy bars? Performance enhancing drugs? Obviously, it has something to do with energy, but you need to be a bit more specific.) Where is the extra energy coming from?
Pick a data set for your analysis.
• olympic-dash-vault.txt
Combined gold medal results from the men's hundred meter dash and pole vault for every one of the modern Olympiads.
• olympic-decathlon.txt
Hundred meter dash and pole vault results of the decathlon gold medalists for every Olympiad in which this event was held.
2. pile-driver.txt
A group of students performed an experiment driving nails into a wooden block. They used a pile driver of mass m = 1.1091 kg released at rest from a height h above the block. Before the pile driver fell, the top of the nail was a height s1 above the block. After the pile driver fell, the top of the nail was a height s2 above the block. They repeated the experiment eight times — four times driving the nail with the grain of the wood and four times driving the nail across the grain. For each trial determine…
1. the displacement of the nail when hit by the pile driver
2. the work done by the falling pile driver
3. the average force exerted by the pile driver on the nail
4. the average acceleration of the pile driver while in contact with the nail
5. the speed of the pile driver on impact
6. the duration of each impact (in milliseconds), and
7. the power of each impact (in kilowatts)
Compile your results in tables like those below.
Nailing with the grain
h
(m)
s1
(m)
s2
(m)
s
(m)
W
(J)
F
(N)
a
(m/s2)
v
(m/s)
t
(ms)
P
(kW)
0.3060 0.07160 0.06600
0.6115 0.06600 0.05675
0.9180 0.05675 0.04435
1.2220 0.04435 0.03060
Nailing across the grain
h
(m)
s1
(m)
s2
(m)
s
(m)
W
(J)
F
(N)
a
(m/s2)
v
(m/s)
t
(ms)
P
(kW)
0.3060 0.06935 0.06670
0.6115 0.06670 0.06095
0.9180 0.06095 0.05370
1.2220 0.05370 0.04640
Lastly, construct a graph of penetration depth vs. average force and…
1. determine the effect that force has on the distance a nail moves for this type of wood.
Source: The hammer falls: A fresh look at the pile driver. Kyle Hathcox and David Ward. The Physics Teacher. Vol. 43 No. 7 (October 2005): 428–431.