Question

If a body be projected vertically upward from the surface of the earth so as to reach a height $nR$ above the surface; the increase in its potential energy is $\left(\frac{n}{n+a}\right)mgR$ ,where $a=$ ?

Answer 1

Potential energy at the surface of earth is

$E=-\frac{GMm}{R}=-gmR$

Potential energy at a height $nR$ above the surface of earth is

$E^{\prime }=-\frac{GMm}{(n+1)R}=-\frac{gmR}{n+1}$

Thus, increment in potential energy is

$\Delta E=E^{\prime }-E=\frac{n}{n+1}mgR$

So, $a=1$.