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

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Solution

For giving minimum speed to escape, the velocity will become zero on reaching infinity.
So, for the particle T.E at infinity $= 0.$
And by energy conservation, total energy at this position is also equal to zero.
$\Rightarrow P E + K E = 0$
$\Rightarrow ⎛ ⎜ ⎜ ⎜ ⎝ \frac{- G M_{1} m}{\frac{d}{2}} - \frac{G M_{2} m}{\frac{d}{2}} ⎞ ⎟ ⎟ ⎟ ⎠ + \frac{1}{2} {m v}^{2} = 0$
$\Rightarrow \frac{- 2 G M_{1}}{d} - \frac{2 G M_{2}}{d} = \frac{- v^{2}}{2}$
$\Rightarrow \frac{v^{2}}{2} = \frac{2 G}{d} (M_{1} + M_{2})$
$\Rightarrow v^{2} = \frac{4 G}{d} (M_{1} + M_{2})$
$\Rightarrow v = 2 \sqrt{\frac{G (M_{1} + M_{2})}{d}} .$
Hence, the answer is $2 \sqrt{\frac{G (M_{1} + M_{2})}{d}} .$

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The masses and radii of the earth and the Moon are M1, R1 and M2, R2, respectively. Their centres are a distance d apart. The minimum speed with which a particle of mass m should be projected from a point midway between the two centres so as to escape to infinity is __________.