Question

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

• A. $\frac{R}{2}$
• B. $\frac{R}{3}$
• C. $\frac{R}{4}$
• D. $\frac{R}{5}$

Answer 1

The correct option is B $\frac{R}{3}$

According to the law of conservation of energy:

Total energy of the body on the surface of earth $=$ Total energy of the body at a height ‘h’ reached by it

$\Rightarrow (P.E+K.E{)}_{Surface}=(P.E+K.E{)}_{MaxHeight}$

$\Rightarrow -\frac{GMm}{R}+\frac{1}{2}m{\left(\frac{V_e}{2}\right)}^2$$=-\frac{GMm}{R+h}+\frac{1}{2}m{\left(0\right)}^2$ ........ (1)

We know that: $V_e=\sqrt{\frac{2GM}{R}}$ ..........(2)

Sunstituting (2) in (1) and solving, we get $h=\frac{R}{3}$