Question

The kinetic energy needed to project a body of mass m at rest from a point P at height R/2 from surface of earth to infinity (R = radius of earth, M = mass of earth) is
1. 2GMm/3R
2. GMm/3R
3. 2GMm/R
4. zero​

Answer 1

The kinetic energy needed to project a body of mass m at rest from a point P at height R/2 from surface of earth to infinity is 2GMm/3R

Therefore, Option (1) is correct.

Explanation:

Given radius of the Earth = R

Mass of the Earth = M

Mass of the object = m

Distance of point P from the centre of the Earth

$R'=R+\frac{R}{2}$

$\implies R'=\frac{3R}{2}$

We know that potential energy at a distance R' from the centre of the Earth is given by

$\boxed{U=-\frac{GMm}{R'}}$

or, $U=-\frac{GMm}{3R/2}$

or, $U=-\frac{2GMm}{3R}$

Potential energy at infinity will be zero

Change in potential energy

$\Delta U=0-(-\frac{2GMm}{3R}$

$\implies \Delta U=\frac{2GMm}{3R}$

This change in potential energy will be equal to the Kinetic Energy

Therefore,

The Kinetic Energy needed to project the object at infinity

$=\frac{2GMm}{3R}$

Hope this answer is helpful.

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