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Question 7.11: A NEAR-EARTH ASTEROID GOAL Use gravitational potential energ...

A NEAR-EARTH ASTEROID

GOAL Use gravitational potential energy to calculate the work done by gravity on a falling object.

PROBLEM An asteroid with mass m=1.00 \times 10^9 \mathrm{~kg} comes from deep space, effectively from infinity, and falls toward Earth. (a) Find the change in potential energy when it reaches a point 4.00 \times 10^8 \mathrm{~m} from the center of the Earth (just beyond the orbital radius of the Moon). In addition, find the work done by the force of gravity. (b) Calculate the asteroid’s speed at that point, assuming it was initially at rest when it was arbitrarily far away. (c) How much work would have to be done on the asteroid by some other agent so the asteroid would be traveling at only half the speed found in (b) at the same point?

STRATEGY Part (a) requires simple substitution into the definition of gravitational potential energy. To find the work done by the force of gravity, recall that the work done on an object by a conservative force is just the negative of the change in potential energy. Part (b) can be solved with conservation of energy, and part (c) is an application of the work-energy theorem.

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