10. Conservation of Energy

By how much does the gravitational potential energy of a 58-kg pole vaulter change if her center of mass rises 4.0 m during the jump?

A sphere of radius r₁ has a concentric spherical cavity of radius r₂ (Fig. 8–42). Assume this spherical shell of thickness r₁ - r₂ is uniform and has a total mass M. Show that the gravitational potential energy of a mass m at a distance r from the center of the shell ( r > r₁) is given by U = - (GmM/r).

The free-fall acceleration on a large asteroid, in the vacuum of space, is 0.15 m/s². A spacecraft hovering 500 m above the surface drops a 25 kg payload wrapped in a padded jacket. What is the payload's impact speed?

A binary star system has two stars, each with the same mass as our sun, separated by 1.0 ✕ 10¹² m. A comet is very far away and essentially at rest. Slowly but surely, gravity pulls the comet toward the stars. Suppose the comet travels along a trajectory that passes through the midpoint between the two stars. What is the comet's speed at the midpoint?