Upon what basic quantity does kinetic energy depend? Upon what basic quantity does potential energy depend?

Can kinetic energy ever be negative? Can gravitational potential energy ever be negative? For each, give a plausible reason for your answer without making use of any equations.

A roller-coaster car rolls down a frictionless track, reaching speed $v_0$ at the bottom. If you want the car to go twice as fast at the bottom, by what factor must you increase the height of the track? Explain.

The three balls in FIGURE Q10.4, which have equal masses, are fired with equal speeds from the same height above the ground. Rank in order, from largest to smallest, their speeds $v_{\text{A}},v_{\text{B}}$, and $v_{\text{C}}$ as they hit the ground. Explain.

FIGURE Q10.4

Rank in order, from most to least, the elastic potential energy $(U_{\text{Sp}}){}_{\text{A}}$ to $(U_{\text{Sp}}){}_{\text{D}}$ stored in the springs of FIGURE Q10.5. Explain.

FIGURE Q10.5

A spring is compressed 1.0 cm. How far must you compress a spring with twice the spring constant to store the same amount of energy?

A spring gun shoots out a plastic ball at speed $v_0$. The spring is then compressed twice the distance it was on the first shot. By what factor is the ball’s speed increased? Explain.

A particle with the potential energy shown in FIGURE Q10.8 is moving to the right at $x=5\text{m}$ with total energy $E$.

1. At what value or values of $x$ is this particle’s speed a maximum?

2. Does this particle have a turning point or points in the range of $x$ covered by the graph? If so, where?

3. If $E$ is changed appropriately, could the particle remain at rest at any point or points in the range of $x$ covered by the graph? If so, where?

FIGURE Q10.8

A compressed spring launches a block up an incline. Which objects should be included within the system in order to make an energy analysis as easy as possible?

A process occurs in which a system’s potential energy decreases while the system does work on the environment. Does the system’s kinetic energy increase, decrease, or stay the same? Or is there not enough information to tell? Explain.

A process occurs in which a system’s potential energy increases while the environment does work on the system. Does the system’s kinetic energy increase, decrease, or stay the same? Or is there not enough information to tell? Explain.

FIGURE Q10.12 is the energy bar chart for a firefighter sliding down a fire pole from the second floor to the ground. Let the system consist of the firefighter, the pole, and the earth. What are the bar heights of $W_{\text{ext}}$ and $K_{\text{f}}?$

FIGURE Q10.12

1. If the force on a particle at some point in space is zero, must its potential energy also be zero at that point? Explain.

2. If the potential energy of a particle at some point in space is zero, must the force on it also be zero at that point? Explain.