Irodov Problem 1.229

When a body is in a parabolic path around a planet its total energy (kinetic energy + potential energy) is exactly equal to 0. Bodies in a gravitational field follow conic sections (ellipse, parabola, hyperbola, circle etc.). When the total energy of the body is negative the body cannot escape the gravitational field of the planet and follows a circle or an elliptical path. When its total energy is positive it have achieved escape velocity and will follow a hyperbolic path. When its total energy is exactly 0, it has just enough energy to escape the planet's gravitational force and it follows a parabolic path.

As the problem says that the rocket was following a parabolic path, its total energy must be zero. Since the parabolic path is tangent to moon at the surface, if the radius of the moon is R, then its velocity v1 when it reaches the closest approach will be,

Now the rocket brakes and falls into a circular orbit. As calculated in problem 1.229, its velocity in the circular orbit will be,


The increase in velocity will be v2 - v1 given by,