A space-ship is launched into a circular orbit close to ther earth's surface. What additional speed should now be imparted to the spaceship so that orbit to overcome the gravitational pull of the earth.

Explanation

Let `Delta K` be the additional kinetic energy imparted to the spaceshi[ to overcome the gravitation pull then `Delta K=-` (total energy of spaceship) `=(GMm)/(2R)`
Total kinetic energy `=(GMm)/(2R) +Delta K= (GMm)/(2R)+(GMm)/(2R)=(GMm)/R`, then `1/2 mv_(2)^(2) =(GMm)/R implies v_(2) = sqrt((2GM)/R)`
But `v_(1)=sqrt((GM)/R)`. So additional velocity required `=v_(2)-v_(1)=sqrt((2GM)/R)-sqrt((GM)/R)=(sqrt(2)-1) sqrt((GM)/R)`
Alternate solution :
Additional velocity = Escape velocity - Orbital velocity
`=v_(es)-v_(0)`
`=sqrt((2GM)/R)-sqrt((GM)/R)`
`=(sqrt(2)-1) sqrt((GM)/R)`