ESCAPE VELOCITY

DEFINITION

Escape velocity is defined as the minimum speed to be given to an object on the surface of a planet if it is to escape from the gravitational pull or field of that planet.

This assumes the object will follow a ballistic trajectory in its escape from the planet. It also implies there is no additional propulsion and that the planet has no atmosphere to impose a frictional drag force during the escape trajectory. Even in the presence of an atmosphere we still talk about an escape velocity as if the planet's atmosphere is removed.

The symbol for the escape velocity is v_e and it varies with the mass and the diameter of the planet.

JULES VERNE

In 1865 the author Jules Verne wrote a novel entitled "From the Earth to the Moon" in which he tells of the Baltimore Gun Club's efforts to build a giant canon with which to launch three people to the Moon.

An artist's impression of the capsule being launched from the mouth of the canon

Despite that fact that the acceleration would kill all the people inside the canon, and the fact that the Earth's atmosphere would impose a large drag force and seriously heat the capsule, this vision comes close to the idea of escape velocity - a speed necessary to give an object on the Earth's surface if it is to entirely escape the gravitational pull of the Earth. No additional propulsion is necessary.

GRAVITATIONAL POTENTIAL ENERGY

Any body with mass M has a graviational field around it. This is defined as the force per unit mass experienced by a small mass m at a distance r from the centre of the body, as long as the distance r is greater then the radius R of the planet. This graviational force is given by:

F_g = G M m / r²

The gravitational field is then:

g = F / m = G M / r²

PE_grav = ∫ [ G M m / r² ] dr

= - G M m / r

where the constant of integration is set to zero to put the zero reference level for the PE at infinity. As the two bodies move closer to one another the PE decreases to larger negative values (see graph below).

ESCAPE VELOCITY CALCULATION

To overcome the potential energy on the surface of the planet we have to supply kinetic energy to the mass m.

KE = 1/2 m v_e²

The equation we have to solve is then:

KE + PE = 0

1/2 m v_e² - G M m / r = 0

which has the solution:

v_e = √ ( 2 G M / r_p )

which is the desired escape velocity.

ESCAPE VELOCITIES FOR SOLAR SYSTEM BODIES

Australian Space Academy