JEE Main · 2021 · 31 Aug 2021 · Shift 1 · Gravitation

The masses and radii of the earth and moon are (

M_{1}, R_{1}

) and (

M_{2}, R_{2}

) respectively. Their centres are at a distance '

r

' apart. Find the minimum escape velocity for a particle of mass '

m

' to be projected from the middle of these two masses:

Answer: (B)

Solution

At the midpoint each mass lies a distance

r /2

away, so the particle's potential energy is

U = - \frac{G M _{1} m}{r /2} - \frac{G M _{2} m}{r /2} = - \frac{2 G ( M _{1} + M _{2} ) m}{r}

. Escape requires the kinetic energy to match the depth of this well:

\frac{1}{2} m V^{2} = \frac{2 G ( M _{1} + M _{2} ) m}{r}

. Solving,

V = \frac{4 G ( M _{1} + M _{2} )}{r} .

JEE Main · 2021 · 31 Aug 2021 · Shift 1 · Single correct option