Question

The kinetic energy required to project a body of mass $m$ from earth's surface (radius $R$) to just escape from earth's atmosphere

• A. $\frac{mgR}{2}$
• B. $2mgR$
• C. $mgR$
• D. $\frac{mgR}{4}$

Answer 1

The correct option is C $mgR$

Given that,
Radius of the earth $=R$
Mass of the body $=m$
Let the mass of the earth is $M$

The potential energy of the earth and body of mass $m$ at the surface of the earth is
$U_i=-\frac{GMm}{R}$

Let the initial kinetic energy needed is $K{E}_i$

The potential energy of the system at infinity, $U_f=0$

The final kinetic energy, $K{E}_f=0$, as the body will just escape the earth's gravity.

Applying conservation of mechanical energy between the surface of earth and infinity:
$U_i+K{E}_i=U_f+K{E}_f$

$\Rightarrow \frac{-GMm}{R}+K{E}_i=0+0$
$K{E}_i=\frac{GMm}{R}=mgR$ ($g=\frac{GM}{R^2}$)

Alternate:

Escape velocity on earth, $v_e=\sqrt{2g{r}_e}$

Kinetic energy required, $K.E=\frac{1}{2}m{v}_e^2=\frac{1}{2}m\times \left(2gR\right)=mgR$

Hence, the correct option is (c)

Why this question? To help students understand the conservation of mechanical energy in planet-satellite systems. It is a very important concept and used frequently in problems.