D.1.13 (HL)—Escape speed

Define escape speed

Escape speed is the minimum speed an object must have at a point—usually the surface of a planet—to reach infinity with zero remaining speed, assuming no air resistance and no other significant gravitational fields. It is not the speed needed to enter a circular orbit.

Derive the model

At the limiting escape condition, initial kinetic energy supplies the increase in potential energy from $-GMm/r$ to zero at infinity. The escaping object’s mass cancels.

Worked example — Earth

With $M=6.0\times10^{24}\,\mathrm{kg}$ and $r=6.4\times10^6\,\mathrm{m}$, $v_{\mathrm{esc}}=\sqrt{2(6.67\times10^{-11})(6.0\times10^{24})/(6.4\times10^6)}=1.1\times10^4\,\mathrm{m\,s^{-1}}$, about $11\,\mathrm{km\,s^{-1}}$. This is the minimum no-drag speed for zero speed at infinity.

Read the scaling

Escape speed increases with the square root of source mass and decreases with the square root of distance from its centre. For bodies with the same density, $M\propto R^3$, so at the surface $v_{\rm esc}\propto R$. Always use the source centre-to-point distance $r$.

Common trap

Do not use the circular-orbit speed $\sqrt{GM/r}$ or say that escape means “overcoming gravity” at a finite boundary. The limiting condition is reaching infinity with zero final speed.