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Q.
A body is thrown with a velocity equal to $n$ times the escape velocity $\left(v_{e}\right)$. Velocity of the body at a large distance away will be
Gravitation
Solution:
At large distance potential energy $=0$
Conserving mechanical energy at surface of earth and large distance from earth,
$\frac{1}{2} m\left(n v_{e}\right)^{2}-\frac{G M m}{R}=\frac{1}{2} m v^{2}$
Also, $v_{e}=\sqrt{\frac{2 G M}{R}}$
$\Rightarrow \frac{2 G M m}{R}\left(n^{2}-1\right)=m v^{2}$
$\Rightarrow v=\sqrt{\frac{2 G M}{R}}\left(n^{2}-1\right)^{1 / 2}$
$\Rightarrow v=v_{e} \sqrt{n^{2}-1}$