A body is projected vertically upwards from the surface of the earth with a velocity sufficient to carry it to infinity. The time taken by it to reach a height of three times the radius of the earth is (acceleration due to gravity =9.8 ~ms ⁻² and radius of the earth =6400 ~km )

Step-by-step solution

If at a distance r from the centre of the earth, the body has velocity v , then by conservation of mechanical energy ( TE )_ at weight =( TE )_ at surface 1 2 m v^2+ (- G M m r )= 1 2 m v_e^2+ (- G M m R ) where, v_e= escape velocity. aligned v^2=2 g R+2 g R ( R r -1 ) aligned Integrating Eq. (i), we get gathered ₀^t d t= 1 R 2 g _R^ R+h r^ 1 / 2 d r t= 2 3 1 R 2 g [(R+h)^ 3 / 2 -R^ 3 / 2 ] t= 1 3 2 R g [ (1+ h R )^ 3 / 2 -1 ] gathered According to question, h=3 R aligned & t= 1 3 2 R g [ (1+ 3 R R )^ 3 / 2 -1 ] &