First slide

Gravitational potential and potential energy

Question

The masses and radii of the earth and moon are $M_{1}, R_{1}$ and $M_{2}, R_{2}$ 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

Moderate

A

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

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

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

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

Solution

Gravitational potential energy at mid point
$U = U_{1} + U_{2} = \frac{- G m M_{1}}{d / 2} + \frac{- G m M_{2}}{d / 2} = \frac{- G m}{d / 2} (M_{1} + M_{2}) = \frac{- 2 G m}{d} (M_{1} + M_{2})$
[m = mass of particle]
So, for projecting particle from mid point to infinity
$K E = | P E |$
$\Rightarrow \frac{1}{2} m v^{2} = \frac{2 G m}{d} (M_{1} + M_{2})$ $\Rightarrow v = 2 \sqrt{\frac{G (M_{1} + M_{2})}{d}}$

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