Question 381: Gravitation

Question

A body is projected vertically from the surface of the earth of radius ' $R$ ' with velocity equal to half of the escape velocity. The maximum height reached by the body is

Accepted Answer

According to the law of conservation of energy,
$$\begin{aligned}
& (	ext { Total energy) })_{	ext {surtace }}=(	ext { Total energy })_{	ext {(max height) }} \
& \Rightarrow \quad(K E+P E)_{	ext {surtace }}=(K E+P E)_{\max 	ext { height }} \
& \Rightarrow \frac{1}{2} m v^2+\left(-\frac{G M m}{R}ight)=0+\left(-\frac{G M m}{R+h}ight)
\end{aligned}$$
Given, $\quad v=\frac{1}{2} v_e=\frac{1}{2} \sqrt{\frac{2 G M}{R}}$
$\Rightarrow \frac{1}{2} m\left[\frac{1}{4} \cdot \frac{2 G M}{R}ight]-\frac{G M m}{R}=-\frac{G M m}{R+h}$
$\Rightarrow \quad \frac{1}{4} \frac{G M m}{R}-\frac{G M m}{R}=-\frac{G M m}{R+h}$
$\Rightarrow \quad \frac{1-4}{4 R}=\frac{1}{R+h} \Rightarrow 3 R+3 h=4 R$
$\Rightarrow \quad h=\frac{R}{3}$
So, option (b) is the correct answer.

Answer

$\frac{R}{5}$

Answer

$\frac{R}{3}$

Answer

$\frac{R}{2}$

Answer

$\frac{R}{4}$

Gravitation

A body is projected vertically from the surface of the earth of radius ' R ' with velocity equal to half of the escape velocity. The maximum height reached by the body is

Choose the correct answer: