Question

A space-ship is launched into a circular orbit close to the Earth,s surface. What additional speed should now be imparted to the spaceship so that it overcome the gravitational pull of the Earth.
Take kinetic energy of the space-ship,
K = total energy of space-ship `= (GMm)/(2R)`
where m = mass of space-ship, M= mass of the earth and R = radius of the earth.

Answer 1

Let `Delta K` be the additional kinetic energy imparted to the spaceship to overcome the gravitation pull then
`Delta K`= - (Total energy of spaceship) = `(GMm)/(2R)`
Total kinetic energy `= (GMm)/(2R)+DeltaK=(GMm)/(2R)+(GMm)/(2R)=(GMm)/(R)`
then `(1)/(2)mv_(2)^(2)=(GMm)/(R)rArr v_(2)=sqrt((2GM)/(R))`
But `v_(1)=sqrt((GM)/(R))`
So Additional velocity
`=v_(2)-v_(1)=sqrt((2GM)/(R))-sqrt((GM)/(R))=(sqrt(2)-1)sqrt((GM)/(R))`.