Show that the velocity of an object released from rest at infinity reaches the earth with a velocity equal to the escape velocity.

Explanation

By the principle of conservation of energy for the object,
TE at infinity = TE at the surfece of the earth
(KE+PE) at infiniy (KE+PE) at the surface of the earth.
As the body is released from rest at infinity, its KE at inifinity is zero and its PE at infinity is also zero. If v is its velocity on reaching the earth its KE becomes equal to `1/2mv^2` and PE=`(GMm)/R`
`therefore 0+0=1/2mv^2 -(GMm)/R , 1/2v^2=(Gm)/R rArr`
The velocity of the object
`V=sqrt((2GM)/R)= V_e` .