Escape velocity

Escape velocity is the minimum speed needed to move infinitely far away from a celestial body, starting from its surface. “Infinitely far” means the body never falls back: in the limiting case it arrives infinitely far away with velocity tending to zero. This condition translates into a total energy of exactly zero.

Escape velocity

Imposing $E_\text{tot} = 0$ (the limit for escape): $\tfrac{1}{2}m v_\text{fuga}^2 - \frac{GMm}{R} = 0 \;\Rightarrow\; \ev{v_\text{fuga} = \sqrt{\frac{2GM}{R}}}$

Key formula

$v_\text{fuga} = \sqrt{\frac{2GM}{R}}$ Independent of the mass $m$ of the projectile: a stone and a spacecraft have the same escape velocity.

The fact that $m$ cancels out is remarkable and has a deep root: the same mass appears in both the kinetic energy and the potential energy, so it simplifies away. It is the same reason why all bodies fall with the same acceleration $g$.

Escape velocity from Earth

With $M_T = 6\cdot 10^{24}\;\text{kg}$ and $R_T = 6{,}4\cdot 10^6\;\text{m}$: $v_\text{fuga} = \sqrt{\frac{2\cdot 6{,}67\cdot 10^{-11}\cdot 6\cdot 10^{24}}{6{,}4\cdot 10^6}} \approx 11{,}2\;\text{km/s}$ By comparison, the orbital velocity to stay in low orbit (at $h = 400\;\text{km}$) is “only” $7{,}7\;\text{km/s}$: escaping costs more than orbiting.

Links

Topics: Gravitazione Concepts: Velocità di fuga · Conservazione dell’energia meccanica Skills: Conservazione dell’energia

Related exercises: Velocità di fuga dalla Terra (energia) · Vero o falso su energia e orbite · Dimostrazione della velocità di fuga