Escape Speed Depends on Mass and Radius, Never on Direction
Escape velocity is the speed at which the total energy becomes zero, so the body just reaches infinity. It depends on the planet's mass and radius only: not on the mass of the escaping body and not on the launch direction.
Motion and force questions become simpler when direction, starting condition, time and the acting forces are marked first.
The formula and its forms
From (1/2) m v² = GMm/R, vₑ = sqrt(2GM/R) = sqrt(2 g R) = R sqrt(8πGρ/3). Because it depends on g and R, two bodies with radii in the ratio p and surface gravity in the ratio q have escape speeds in the ratio sqrt(p q). A planet with 100 times the Earth's mass and 4 times its radius has vₑ = 11.2 x sqrt(100/4) = 56 km/s. A planet with three times the radius and three times the density has mass 81 times larger, so vₑ is sqrt(81/3) = 3 sqrt(3) times larger. Escape speed is the same for a body thrown at 45° as for one thrown vertically, 11.2 km/s for the Earth.
Escape energy from a height
To escape from a distance r from the centre the minimum energy is GMm/r. At 4 Rₑ above the surface the distance is 5 Rₑ, so the energy is GMm/(5 Rₑ) = m g Rₑ / 5, using GM = g Rₑ². The same ideas explain why a space journey uses most of its fuel climbing out of the Earth's gravity on the outward trip.
Black holes
A black hole is a body whose escape speed exceeds the speed of light, so its radius is at most 2GM/c². For the Earth, 2 x 6.67 x 10⁻¹¹ x 5.98 x 10²⁴ / (3 x 10⁸)² is about 9 x 10⁻³ m, so the Earth would have to be compressed to roughly a centimetre.
OlympIQ exam lens
- ✓vₑ = sqrt(2GM/R) = sqrt(2gR); independent of the escaping body's mass and of the launch angle
- ✓Ratios: vₑ is proportional to sqrt(M/R) and to R sqrt(ρ)
- ✓Escape energy at distance r is GMm/r
- ✓Black-hole radius is 2GM/c²; for the Earth about 1 cm
Do not assume a body thrown at an angle needs a different escape speed. Energy, not direction, decides escape, so the speed is the same, 11.2 km/s for the Earth, however it is launched (as long as it does not hit the ground).
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