Question

A particle of mass m is kept at rest at a height 3r

Answer 1

your question is incomplete. A complete question is ----> A particle of mass m is kept at rest at a height 3R from the surface of earth, where R is radius of earth and M is mass of earth. The minimum speed with which it should be projected, so that it does not return back, is (g is acceleration due to gravity on the surface of earth) ?

solve :- we must use the conservation law of energy. we know, the sum of potential energy and its kinetic energy remains constant.
$P.E_i+K.E_i=P.E_f+K.E_f$

as initial velocity of particle is zero so, initial kinetic energy of particle must be zero.
e.g., $K.E_i=0$
potential energy, $P.E_i=-\frac{GMm}{3R}$

Let final velocity of particle is v
then, final kinetic energy of particle , $K.E_f=\frac{1}{2}mv^2$

final gravitational potential energy, $P.E_f=-\frac{GMm}{R}$

now, $-\frac{GMm}{3R}=-\frac{GMm}{R}+\frac{1}{2}mv^2$

or, $\frac{4GM}{3R}=v^2$

hence, $v=2\sqrt{\frac{GM}{3R}}$

The minimum speed with which it should be projected, so that it does not return back, is $v=2\sqrt{\frac{GM}{3R}}$

Answer 2

Answer:√(MG/2)

Explanation:

We should find the escape velocity:

v=√[2MG/(R+h)]

h=3R

v=√[2MG/(R+3R)]

=√(2MG/4R)

=√(MG/2R)