Escape Velocity Formula
Calculate the minimum speed needed to escape a planet or star's gravitational pull.
Essential for space travel calculations.
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The Formula
v = √(2GM / r)
Escape velocity is the minimum speed an object needs to break free from a body's gravitational pull without further propulsion. It depends only on the mass and radius of the body — not on the mass of the escaping object.
Variables
| Symbol | Meaning |
|---|---|
| v | Escape velocity (m/s) |
| G | Gravitational constant (6.674 × 10⁻¹¹ N⋅m²/kg²) |
| M | Mass of the body being escaped from (kg) |
| r | Distance from the center of the body (meters) |
Example 1
Find Earth's escape velocity
M = 5.972 × 10²⁴ kg, r = 6.371 × 10⁶ m
v = √(2 × 6.674 × 10⁻¹¹ × 5.972 × 10²⁴ / 6.371 × 10⁶)
v = √(1.251 × 10⁸)
v ≈ 11,186 m/s ≈ 11.2 km/s (about 40,270 km/h)
Example 2
Find the Moon's escape velocity
M = 7.342 × 10²² kg, r = 1.737 × 10⁶ m
v = √(2 × 6.674 × 10⁻¹¹ × 7.342 × 10²² / 1.737 × 10⁶)
v ≈ 2,376 m/s ≈ 2.38 km/s
When to Use It
Use the escape velocity formula when:
- Planning spacecraft launches and trajectories
- Comparing the gravitational strength of different planets or moons
- Understanding why some bodies retain atmospheres and others do not
- Calculating whether an object will remain in orbit or fly away
Key Notes
- Formula: v_e = √(2GM/R): G = 6.674 × 10⁻¹¹ N·m²/kg², M is the body's mass, R is the radius. Earth's escape velocity ≈ 11.2 km/s; Moon ≈ 2.4 km/s; Sun ≈ 617 km/s.
- Independent of the escaping object's mass: The escape velocity is the same for a 1 kg satellite and a 1,000 kg spacecraft, because the gravitational PE and KE both scale with mass and cancel out.
- Requires no further thrust: Escape velocity is the minimum speed needed at the surface to escape with no additional propulsion. Real rockets burn fuel continuously and don't need to achieve escape velocity all at once.
- Atmospheric drag is excluded: The formula assumes a vacuum. In practice, rockets must overcome atmospheric drag, which increases the effective required speed, especially in the dense lower atmosphere.
- Black holes and light: If a body's escape velocity exceeds the speed of light (c ≈ 3×10⁸ m/s), even light cannot escape — this defines the event horizon of a black hole, where R = 2GM/c² (the Schwarzschild radius).