Question

The kinetic energy needed to project a body of mass m from the earth' surface to infinity is
(a) $\frac{1}{4}mgR$

(b) $\frac{1}{2}mgR$

(c) $mgR$

(d) $2mgR$.

Answer 1

(c) $mgR$

The kinetic energy needed to project a body of mass m from the Earth's surface to infinity is equal to the negative of the change in potential energy of the body.

i.e., kinetic energy = $-$(final potential energy $-$ initial potential energy)
$$\begin{gathered} \Rightarrow K=-\left(\frac{GMm}{R\text{'}}-\frac{GMm}{R}\right) \\ \Rightarrow K=-\left(\frac{GMm}{\infty }-\frac{GMm}{R}\right)=\frac{GMm}{R} \\ \Rightarrow K=mR\times \left(\frac{GM}{R^2}\right) \\ \Rightarrow K=mRg\left[\therefore g=\frac{GM}{R^2}\right] \end{gathered}$$