Question

The kinetic energy needed to project a body of mass m from the earth' surface to infinity is

Options

• \[\frac{1}{4}\]mgR

• \[\frac{1}{2}\]mgR

• mgR

• 2 mgR

Answer 1

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)

\[\Rightarrow K = - \left( \frac{GMm}{R'} - \frac{GMm}{R} \right)\]

\[ \Rightarrow K = - \left( \frac{GMm}{\infty} - \frac{GMm}{R} \right) = \frac{GMm}{R}\]

\[ \Rightarrow K = mR \times \left( \frac{GM}{R^2} \right)\]

\[ \Rightarrow K = mRg \left[ \therefore g = \frac{GM}{R^2} \right]\]