An artificial satellite is moving in a circular orbit around the earth with a speed of equal to half the magnitude of escape velocity from earth.
(i). Determine the height of the satellite above the earth's surface
(ii). If the satellite is stopped suddenly in its orbit and allowed to fall freely on the earth. Find the speed with it hits and surface of earth. Given `M="mass of earth & R "="Radius of earth"

Explanation

(a) Let height above the earth's surface = h then
`v_("orbital")= sqrt((GM)/(R+h))=1/2 v_(e)=1/2 sqrt((2GM)/R)=sqrt((GM)/(2R)) implies R+h=2R implies h=R`
(b) If the satellite is stopped suddenly then it total energy `E_(1)= - (GMm)/(2R)`
Let its speed be v when it hits the earth's surface then its total energy on earth surface `E_(2)=- (GMm)/R +1/2 mv^(2)`
conservation law for mechanical energy yields `E_(1)=E_(2)implies (-GMm)/(2R) = - (GMm)/R +1/2 mv^(2) implies v=sqrt((GM)/R)`