Escape Velocity

Escape velocity is defined as the minimum speed with which an object must be launched from a planet's surface, straight upward (with no further propulsion after launch, and ignoring air resistance), so that it can travel arbitrarily far away and just barely reach infinity -- with, at that limiting case, zero velocity left over on arrival.

This condition is most cleanly expressed using energy conservation. At launch, from the surface (distance $R$ from the centre), the object of mass $m$ has kinetic energy $\tfrac{1}{2}mv_e^2$ and gravitational potential energy $U(R) = -GMm/R$ (Section 7.8); at the limiting case of "just barely" reaching infinity, both its kinetic energy AND its potential energy are zero (potential energy is defined to be zero at infinity, and the object has no leftover speed there either). Since no other force does work on it along the way, total mechanical energy is conserved between these two states:

Solving for $v_e$ (the one factor of $m$ cancelling out of both terms):

using $g = GM/R^2$ from Section 7.5 for the second, often more convenient, form. For the Earth, taking $g = 9.8\ \text{m/s}^2$ and $R = 6.4 \times 10^6\ \text{m}$,

A few points about escape velocity are worth being clear about. First, it does NOT depend on the mass of the escaping object itself, nor on the direction it is launched (the same expression, $\sqrt{2GM/R}$, applies whether the launch is vertical or at any other angle, though the actual trajectory taken naturally differs) -- it depends only on the planet's own mass $M$ and radius $R$ (or, equivalently, on that planet's own surface value of $g$ and its radius $R$). Second, "escaping" here does not mean escaping ALL gravitational influence in the universe -- only the influence of the one particular body being launched from; an object launched from the Earth at escape velocity is still bound to (and orbits) the Sun. …