The second cosmic velocity (escape velocity) is the smallest speed that must be given to a body at a planet's surface for it to escape the planet's gravitational field completely and move away without limit — so that it never returns to the surface.
Symbols
vII
second cosmic velocity (escape velocity)
[m/s]
G
gravitational constant (≈6.67⋅10−11)
[N·m²/kg²]
M
mass of the central body (the planet)
[kg]
R
radius of the planet (distance from its centre)
[m]
Where it comes from
We derive it from the conservation of energy. For a body to "escape", its kinetic energy must be enough to overcome the gravitational attraction: 21mv2=RGMm. The body's mass m cancels, leaving v2=R2GM, that is vII=R2GM. Comparing with the first cosmic velocity gives a simple relation: vII=2⋅vI — the escape velocity is 2 (≈1.41) times the first.
When to use it
When asking what speed lets a body leave the planet for good (escape the gravitational field), not merely enter orbit. For Earth vII≈11.2 km/s. The relation vII=2⋅vI always holds.
Example
Find the second cosmic velocity for Earth.
SubstitutionM=5.97⋅1024 kg, R=6.37⋅106 m, G=6.67⋅10−11.
ResultvII=6.37⋅1062⋅6.67⋅10−11⋅5.97⋅1024≈1.12⋅104 m/s ≈11.2 km/s. (Check: 2⋅7.9 km/s ≈11.2 km/s.)