Question

The kinetic energy needed to project a body of mass $m$ from earth's surface $($ radius $R$ $)$ to infinity is

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

Answer 1

The correct option is C $mgR$

For the body to escape Earth's gravitation and reach infinity, the initial kinetic energy has to be just higher than the Gravitational potential energy of the body on the surface.
Total energy of the body just after projecting is:
$E=\frac{1}{2}m{v}^2-\frac{GMm}{R}=\frac{1}{2}m{v}^2-mgR$
To escape Earth's gravitation, this energy has to be positive:
$\Rightarrow \frac{1}{2}m{v}^2-mgR\ge 0\Rightarrow K.E\ge mgR$
So, the minimum Kinetic energy required is $mgR$.