A spaceship is launched into a circular orbit close to the Earth's surface. What additional velocity has to be imparted to the spaceship to overcome the gravitational pull?

Explanation

Let `DeltaK` be the additional kinetic energy imparted to the spaceship to overcome the gravitation pull then `DeltaK=-("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)impliesv_(2)=sqrt((2"GM")/(R))`
But `v_(1)=sqrt(("GM")/("R"))-sqrt(("GM")/("R"))=(sqrt(2)-1)sqrt(("GM")/("R"))`
Alternate Solution :
Additional velocity = Escape velocity-Orbital velocity
`=v_("es")-v_(0)=sqrt((2"GM")/("R"))-sqrt(("GM")/("R"))=(sqrt(2)-1)sqrt(("GM")/("R"))`