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Kinetic Energy

The masses an...

Question

The masses and radii of the earth and moon are $M_{1}, R_{1}$ and $M_{2}, R_{2}$ respectively. Their centers are at distance d apart. The minimum speed with which a particle of mass m should be projected from a point midway the two centers so as to escape to infinity is:

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Solution

The law of conservation of energy:
T + U = const
where $T = \frac{m v^{2}}{2}$ is kinetic energy, m - mass, v - speed, $U = - \frac{G m M}{r}$ is potential energy, r is the distance to the center of second body.
Therefore:
$- \frac{G M_{1} m}{\frac{d}{2}} - \frac{G M_{2} m}{\frac{d}{2}} + \frac{m v^{2}}{2} = 0 + 0 + 0$
$v^{2} = \frac{4 G}{d} (M_{1} + M_{2})$
$v = \sqrt{\frac{4 G}{d} (M_{1} + M_{2})}$
$v = \sqrt{\frac{4 G}{d} (M_{1} + M_{2})}$

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