Orbits of planets and satellites (A-level only): Escape velocity.

The idea

Escape velocity is the minimum launch speed needed for a projectile to leave a planet or moon without further propulsion and never fall back. Energy analysis gives the method: at the surface the body has kinetic energy ½mv² and gravitational potential energy −GMm/r; at escape the total energy is zero, so ½mv² = GMm/r. Cancelling m gives v = √(2GM/r), showing escape speed is independent of the escaping mass. For Earth, using G = 6.67 × 10⁻¹¹ N m² kg⁻², M = 5.97 × 10²⁴ kg and r = 6.37 × 10⁶ m gives about 1.12 × 10⁴ m s⁻¹. The same formula applies to any spherical body of mass M and radius r, and the value is a speed, not a fixed direction.

What earns the marks

• States that escape occurs when the total energy (kinetic plus gravitational potential) of the projectile is zero or greater, so it can reach infinity with zero kinetic energy.
• Derives or quotes the energy condition ½mv² = GMm/r and cancels the mass m to show escape speed is independent of the projectile mass.
• Rearranges correctly to v = √(2GM/r) and substitutes G, planetary mass M and distance r from the centre, not the surface altitude.
• Evaluates a numerical escape speed with consistent SI units, for example about 1.1 × 10⁴ m s⁻¹ for Earth, and states the unit m s⁻¹.
• Explains that escape speed is a minimum speed and that a larger launch speed still allows escape, while a smaller speed leads to a bound orbit or return.
• Uses gravitational potential energy as negative, −GMm/r, and recognises that zero total energy is the boundary between bound and unbound motion.

Common mistakes and how to put them right

• Using the surface altitude above the planet instead of the distance from the planet's centre.

  Put it right by adding the planet's radius to the altitude before substituting into v = √(2GM/r).

• Treating escape velocity as dependent on the mass of the escaping object.

  Put it right by cancelling m in the energy equation, which shows the mass cancels and the speed depends only on M and r.

• Forgetting that gravitational potential energy is negative and using +GMm/r.

  Put it right by writing the potential energy as −GMm/r so that the zero total energy condition gives ½mv² = GMm/r.

Exam tips

• Start from the energy condition and show the cancellation of m explicitly; this demonstrates understanding rather than quoting a formula.
• Check that r is measured from the centre of the planet and convert kilometres to metres before calculating.
• Give the final answer to an appropriate number of significant figures and include the unit m s⁻¹, since a speed without a unit is incomplete.