Editing Escape velocity (section)

=== Barycentric escape velocity ===
Escape velocity can either be measured as relative to the other, central body or relative to [[center of mass|center of mass or barycenter]] of the system of bodies. Thus for systems of two bodies, the term ''escape velocity'' can be ambiguous, but it is usually intended to mean the barycentric escape velocity of the less massive body. Escape velocity usually refers to the escape velocity of zero mass [[test particles]]. For zero mass test particles we have that the 'relative to the other' and the 'barycentric' escape velocities are the same, namely <math>v_\text{e} = \sqrt{\frac{2GM}{d}} </math>.

But when we can't neglect the smaller mass (say ''m'') we arrive at slightly different formulas.

Because the system has to obey the [[Momentum#Conservation|law of conservation of momentum]] we see that both the larger and the smaller mass must be accelerated in the gravitational field. Relative to the center of mass the velocity of the larger mass (''v''{{sub|p}}, for planet) can be expressed in terms of the velocity of the smaller mass (''v''{{sub|r}}, for rocket). We get <math>v_p=-\frac{m}{M}v_r</math>.

The 'barycentric' escape velocity now becomes <math>v_r=\sqrt{\frac{2GM^2}{d(M+m)}} \approx \sqrt{\frac{2GM}{d}}</math>, while the 'relative to the other' escape velocity becomes <math> v_r -v_p=\sqrt{\frac{2G(m+M)}{d}} \approx \sqrt{\frac{2GM}{d}}</math>.