Question

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

• A. $v_e\sqrt{n^2-1}$
• B. $v_e\sqrt{n^2+1}$
• C. $v_e\sqrt{1-n^2}$
• D. None of these

Answer 1

Answer: (A)

At large distance potential energy $=0$
Conserving mechanical energy at surface of earth and large distance from earth,
$\frac{1}{2}m{\left(n{v}_e\right)}^2-\frac{GMm}{R}=\frac{1}{2}m{v}^2$
Also, $v_e=\sqrt{\frac{2GM}{R}}$
$\Rightarrow \frac{2GMm}{R}\left(n^2-1\right)=m{v}^2$
$\Rightarrow v=\sqrt{\frac{2GM}{R}}{\left(n^2-1\right)}^{1/2}$
$\Rightarrow v=v_e\sqrt{n^2-1}$