Question

A body is released from a height equal to radius(R) of the earth. The velocity of body when it strikes the surface of the earth will be :

• A. $\sqrt{gR}$
• B. $\sqrt{2gR}$
• C. $gR$
• D. $2\sqrt{gR}$

Answer 1

The correct option is C $gR$

The potential energy of a mass m
at a distance r
from the center of the earth (assumed to be a spherically symmetric object of mass M
and radius R

) is

$V_r=GMmr$

Initially the mass is at rest at $r=2R$

its total energy is

$V\left(2R\right)=GMm2R$

Thus, from the law of conversation of energy its speed v

when it hits the earth (at r=R) is given by

$12m{v}_{2}GMmR=GMm2R$

$12m{v}_2=GMm\left(R_1{R}_2\right)=GMmR2$

$V_2=GMR$

Now, the force on the mass at the surface of the earth is given by

$mg=GMmR2g=GM{R}_2$

Thus

$V_2=gRv=gR$