The mass and radius of the earth and moon are $M, R$ and $m, r$ respectively. The distance between their…
The mass and radius of the earth and moon are $M, R$ and $m, r$ respectively. The distance between their centers is $d$. The minimum velocity with which a particle of mass $m_0$ should be projected from the midpoint between them so that it could reach infinity is
$2 \sqrt{\frac{G}{d}(M+m)}$
$2 \sqrt{\frac{G m}{d}(M+m)}$
$2 \sqrt{\frac{2 G}{d}(M+m)}$
$2 \sqrt{\frac{G m(M+m)}{d(R+r)}}$
Solution
The potential energy at the midpoint is $\left(-\frac{G m m_0}{d / 2}-\frac{G M m_0}{d / 2}\right)$
And suppose $\left(\frac{m_0 V^2}{2}\right)$ kinetic energy is required to just send the particle to infinity.
On considering energy conservation, i.e., initial total energy is equal to the final total energy:
$\begin{aligned}
& \left(-\frac{G m m_0}{d / 2}-\frac{G M m_0}{d / 2}\right)+\left(\frac{m_0 V^2}{2}\right)=0 \\
& \Rightarrow V^2=\frac{4 G}{d}(M+m) \\
& \Rightarrow V=2 \sqrt{\frac{G(M+m)}{d}}
\end{aligned}$