An object is allowed to fall freely towards the earth from a distance `r ( gt R_e)` from the centre of the earth. Find the speed of the object when it strikes the surface of the earth.

Explanation

Initial speed of body is zero, hence total energy,
`E=(GM_em)/r " " .......(1)`
Suppose mass of body is ` m_1` mass of earth `M_(e)` radius of earth `R_e and v` is the speed of body when it hit the earth.
Total energy of body when body hit the earth,
`E = 1/2 mv^(2) - (GM_(e)m)/R_e " " ..........(2)`
Neglecting air resistance, according to conservation of mechanical energy,
`1/2mv^2-(GM_em)/R_e=(GM_em)/r`
`:.v^2=2GM_e(1/R_e-1/r)`
but `g = (GM_e)/R_e^2`
`:. v^2=2R_e^2g(1/R_e-1/r)` or
`:. v =R_e[2g(1/R_e-1/r)]^(1/2)`