An astronaut takes a ball of mass m from earth to space. He throws the ball into a circular orbit about earth at an altitude of 318.5 ~km . From earth's surface to the orbit, the change in total mechanical energy of the ball is x GM _ e m 21 R _ e . The value of x is (take . R _ e =6370 ~km ) :

Step-by-step solution

aligned & h =318.5 ( R _ e 20 ) & T E _ i = - GM _ e m R _ e & T E _ f = - GM _ e m 2 ( R _ e + h ) = - GM _ e m 2 ( R _ e + R _ e 20 ) & T E _ f = -10 GM _ e m 21 R _ e aligned Change in total mechanical energy aligned & = TE _ f - TE _ i & = GM _ e m Re [1- 10 21 ]= 11 GM _ e m 21 Re aligned