1. Work, Energy, Power

2. ConcepTest 1: Kinetic energy
Car #1 has twice the mass of car #2, but they both have the same kinetic energy. How do their speeds compare?
1. 2 v1 = v2
2. √2 v1 = v2
3. 4 v1 = v2
4. v1 = v2
5. v1 = √2 v2
Answer: 2. Kinetic energy is proportional to the mass and proportional to the square of the velocity. Since the mass of car #1 is twice the mass of car #2, the square of the velocity of car #2 must be twice the square of the velocity of car #1. Hence, the velocity of car #2 must be √2 times the velocity of car #1.

3. ConcepTest 1: Kinetic energy
Car #1 has twice the mass of car #2, but they both have the same kinetic energy. How do their speeds compare?
2. √2 v1 = v2
Kinetic energy is proportional to the mass and proportional to the square of the velocity. Since the mass of car #1 is twice the mass of car #2, the square of the velocity of car #2 must be twice the square of the velocity of car #1. Hence, the velocity of car #2 must be √2 times the velocity of car #1.
Answer: 2. Kinetic energy is proportional to the mass and proportional to the square of the velocity. Since the mass of car #1 is twice the mass of car #2, the square of the velocity of car #2 must be twice the square of the velocity of car #1. Hence, the velocity of car #2 must be √2 times the velocity of car #1.

4. ConcepTest 2: Potential energy
You push a box halfway up a ramp, so that it has potential energy equal to Z. If you push the box all the way up the ramp, the potential energy will be
1. equal to Z.
2. greater than Z, but less than 2 Z.
3. equal to 2 Z.
4. one-half Z.
5. impossible to determine
Answer: 3. The gravitational potential energy is proportional to the mass and to the height. Since the mass is fixed and the height is doubled, the potential energy is also doubled.

5. ConcepTest 2: Potential energy
You push a box halfway up a ramp, so that it has potential energy equal to Z. If you push the box all the way up the ramp, the potential energy will be
3. equal to 2 Z.
The gravitational potential energy is proportional to the mass and to the height. Since the mass is fixed and the height is doubled, the potential energy is also doubled.
Answer: 3. The gravitational potential energy is proportional to the mass and to the height. Since the mass is fixed and the height is doubled, the potential energy is also doubled.

6. ConcepTest 3: Conservation of energy
Two marbles, one twice as heavy as the other, are dropped to the ground from the roof of a building. For these small marbles, air resistance is negligible. Just before hitting the ground, the heavier marble has
1. as much kinetic energy as the lighter one.
2. twice as much kinetic energy as the lighter one.
3. half as much kinetic energy as the lighter one.
4. four times as much kinetic energy as the lighter one.
5. impossible to determine
Answer: 2. Since they start out at the same height, the heavier marble starts out with twice the gravitational potential energy as the lighter one. Since potential energy is proportional to mass, the heavier marble will lose twice the amount of potential energy as the lighter one. Thus, it ends up with twice the kinetic energy.

7. ConcepTest 3: Conservation of energy
Two marbles, one twice as heavy as the other, are dropped to the ground from the roof of a building. For these small marbles, air resistance is negligible. Just before hitting the ground, the heavier marble has
2. twice as much kinetic energy as the lighter one.
Since they start out at the same height, the heavier marble starts out with twice the gravitational potential energy as the lighter one. Since potential energy is proportional to mass, the heavier marble will lose twice the amount of potential energy as the lighter one. Thus, it ends up with twice the kinetic energy.
Answer: 2. Since they start out at the same height, the heavier marble starts out with twice the gravitational potential energy as the lighter one. Since potential energy is proportional to mass, the heavier marble will lose twice the amount of potential energy as the lighter one. Thus, it ends up with twice the kinetic energy.