The potential energy of masses:

$\overline{){\mathbf{U}}{\mathbf{=}}{\mathbf{-}}\frac{\mathbf{G}\mathbf{M}\mathbf{m}}{\mathbf{R}}}$ R is the separation of the masses.

Law of conservation of energy:

$\overline{){{\mathbf{K}}}_{{\mathbf{i}}}{\mathbf{+}}{{\mathbf{U}}}_{{\mathbf{i}}}{\mathbf{+}}{{\mathbf{W}}}_{{\mathbf{nc}}}{\mathbf{=}}{{\mathbf{K}}}_{{\mathbf{f}}}{\mathbf{+}}{{\mathbf{U}}}_{{\mathbf{f}}}}$, where W_{nc} is the work done by non-conservative forces such as gravity.

**Energy of a Spacecraft**

Very far from earth (at R = ∞), 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 theearth. 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.

Part A. Find the speed s_{e} of the spacecraft when it crashes into the earth. Express the speed in terms of M_{e}, R_{e}, and the universal gravitational constant G.

Part B. Now find the spacecraft's speed when its distance from the center of the earth is R = αR_{e} where α ≥ 1. Express the speed in terms of s_{e} and α.

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