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# Speed of a spacecraft as a function of the distance to Earth

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Very far from earth (at R=infinity), a spacecraft has run out of fuel and its kinetic energy is zero. If only the gravitational force of the earth were to act on it (i.e., neglect the forces from the sun and other solar system objects), the spacecraft would eventually crash into the earth. The mass of the earth is M_e and its radius is R_e. Neglect air resistance throughout this problem, since the spacecraft is primarily moving through the near vacuum of space.

Find the spacecraft's speed when its distance from the center of the earth is r.

##### Solution Summary

It is shown how the problem can be solved using conservation of energy.

##### Solution Preview

Potential energy in Earth's gravitational field, W(r) is:

W(r) = -G M_e m/r, (1)

where G is the gravitational constant and ...

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