A body is thrown with a velocity equal to n times escape velocity left(vₑright) . Velocity of body

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7.Gravitation

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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

$v_\theta \sqrt{n^2+1}$

C

$v_e \sqrt{1-n^2}$

D

None of these

Solution

(a)

At large distance potential energy $=0$

Conserving mechanical energy at surface of earth and large distance from earth,

$\frac{1}{2} m\left(n v_e\right)^2-\frac{G M m}{R}=\frac{1}{2} m v^2$

Also, $v_\theta=\sqrt{\frac{2 G M}{R}}$

$\Rightarrow \frac{2 G M m}{R}\left(n^2-1\right)=m v^2$

$\Rightarrow v=\sqrt{\frac{2 G M}{R}}\left(n^2-1\right)^{1 / 2}$

$\Rightarrow v=v_e \sqrt{n^2-1}$

Standard 11

Physics

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