Question

A spaceship is orbiting the Earth around the circular orbit of radius R=2.5Ro, where Ro is the Earth’s radius. What should be the additional velocity V to be supplied to the spaceship in the direction away from the center of the Earth (along the radius of the orbit) so the ship may completely escape the Earth gravity?

Answer 1

here is the picture

Answer 2

I think you would need the mass of the spaceship to determine this, unless i am missing something.

Answer 3

An energy perspective is most useful here. You need sufficient kinetic energy directed radially outward to raise the potential energy of the spacecraft from what it is at R to what it would be at infinity.
\[dU = 0 - \left( - \frac{G M m}{R} \right) = \frac{G M m}{R}\]
Set that equal to the additional kinetic energy:
\[dE_k = \frac{1}{2} m v^2 = \frac{G M m}{R}\]
Now solve for v. G is Newton's constant and M is the mass of the Earth. Fortunately the mass of the spacecraft drops out, so you don't need to know it.