Escape Velocity

Escape velocity is the minimum speed at which an object projected from a planet's surface barely reaches infinity with zero kinetic energy left.

Concept

Setting total energy to zero (just escapes): $\tfrac{1}{2}mv_e^2 - \frac{GMm}{R} = 0$.

$v_e = \sqrt{\frac{2GM}{R}} = \sqrt{2gR}$

For Earth, $v_e \approx 11.2\,\mathrm{km/s}$; for Moon $\approx 2.4\,\mathrm{km/s}$.

Derivation

At infinity KE_∞ ≥ 0 and U(∞)=0 ⇒ total E ≥ 0. Critical case E=0 gives $v_e=\sqrt{2GM/R}$.

Worked Example

Mars: $g_{\rm Mars}=3.71\,\mathrm{m/s^2}$, $R_{\rm Mars}=3389\,\mathrm{km}$. $v_e=\sqrt{2 \cdot 3.71 \cdot 3.389\times 10^6}=5.03\,\mathrm{km/s}$.

Common Confusions

• Escape velocity is INDEPENDENT of mass and direction of projection (gravity is conservative).
• It's a minimum SPEED, not a velocity vector.
• Air resistance ignored — real rockets need higher speeds plus propulsion.

Key Takeaways

• $v_e=\sqrt{2}\,v_{\rm orb}$ for surface-orbit comparison.
• Doesn't depend on the projectile's mass.