Irodov Problem 1.230

This question asks exactly the opposite of problem 1.229. There is a satellite in a circular orbit and now it needs to be imparted some minimal additional velocity to escape. When the satellite is imparted the minimum required velocity to escape its total energy will be 0 and get into a parabolic orbit. So the additional velocity is simply,

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,

Irodov Problem 1.228

When an object is rotating at a distance r in a stable circular orbit around a planet of mass M, its velocity vr is given by,




If the body is orbiting very close to the palnet's surface then its orbital velocity will be,




The total energy (sum of potential energy and kinetic energy) of a body that cannot escape from the gravitational force of a planet is always negative since its negative potential energy exceeds the positive kinetic energy. For any body to be free from the influence of the gravitational field of a planet it needs to have a total positive energy, in other words its kinetic energy must exceed its negative potential energy. Escape velocity is the minimum velocity which is needed by a body to escape the gravitational force of a planet from its surface. The total energy of a body that has escape velocity is 0, since with this minimum kinetic energy it just manages to escape from the planets gravitational force. So if the planets radius is R, then,




Now knowing that,










We can compute the orbital and escape velocities of moon as 1.68 Km/s and 2.377 Km/s. For Earth they are, 7.91 Km/s and 11.186 Km/s.