First slide

Motion of planets and satellites

Question

The masses and radii of the earth and moon arc M₁, R₁ and M₂, R₂, respectively. Their centres are distance d apart. The minimum velocity with which a particle of mass m should be projected from a point midway between their centres so that it escapes to infinity is

Difficult

A

$2 \sqrt{\frac{G}{d} (M_{1} + M_{2})}$
B

$2 \sqrt{\frac{2 G}{d} (M_{1} + M_{2})}$
C

$2 \sqrt{\frac{Gm}{d} (M_{1} + M_{2})}$
D

$2 \sqrt{\frac{Gm (M_{1} + M_{2})}{d (R_{1} + R_{2})}}$

Solution

Gravitational potential at mid point
$V = \frac{- {GM}_{1}}{d / 2} + \frac{- {GM}_{2}}{d / 2}$
$Now, PE = m \times V = \frac{- 2 Gm}{d} (M_{1} + M_{2})$
[m = mass of particle]
So, for projecting particle from mid point to infinity
$\begin{array}{l} KE = | PE | \\ \Rightarrow \frac{1}{2} {mv}^{2} = \frac{2 Gm}{d} (M_{1} + M_{2}) \\ \Rightarrow v = 2 \sqrt{\frac{G (M_{1} + M_{2})}{d}} \end{array}$

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