2001 AIPMT GravitationGravitational Potential Energy Medium Question 5 of 5 in Gravitational Potential Energy

With what velocity should a particle be projected so that its height becomes equal to radius of earth -

A $\left(\dfrac{GM}{R}\right)^{1/2}$
B $\left(\dfrac{8GM}{R}\right)^{1/2}$
C $\left(\dfrac{2GM}{R}\right)^{1/2}$
D $\left(\dfrac{4GM}{R}\right)^{1/2}$

The answer is A.

Solution

Apply energy conservation between the surface (r = R) and the highest point (r = 2R, where the speed is zero):
$-\dfrac{GMm}{R} + \dfrac{1}{2}mv^2 = -\dfrac{GMm}{2R}$
$\dfrac{1}{2}mv^2 = \dfrac{GMm}{R} - \dfrac{GMm}{2R} = \dfrac{GMm}{2R}$
$v = \sqrt{\dfrac{GM}{R}}$
So the particle must be projected with velocity $\left(\dfrac{GM}{R}\right)^{1/2}$.

Concept Energy conservation for large heights

Gravitational Potential Energy in past papers

5 questions from this chapter have appeared across 4 exam years.