Escape Velocity

Even though the gravitational force acts infinitely far, an object can escape the gravitational attraction. The minimum velocity needed for an object to escape from the gravitational influence of a massive body is called escape velocity. It can be calculated by applying conservation of energy and equating the total energy of the object at an initial distance R from the massive body with the energy when the object reaches 'infinity', with a velocity equal 0, presented in the second row in figure 1, from which v_esc can be derived, presented in the third row of the figure 1, which is independent of the object's mass 'm' and true for ballistic motion (no propulsion).

If the body moves with escape velocity but not directly away from the object, it will escape in a curved path (escape orbit) which is parabolic. Above that velocity, the path will be hyperbolic, below the object cannot escape the gravitational influence and will enter a bound orbit. As Newton derived from his law of universal gravitation, the possible trajectories can be described by conic sections.

Kinetic and potential energy equations transition to the escape velocity equation which results in the square root of two times the gravitational constant times the gravitational mass, divided by the distance between the object and the center of the gravitational body.

Figure 1. Equations for derivations of escape velocity.

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Article

Bound and Unbound Trajectories