Time requirement: 50 minutes as an activity for older or mathematically advanced students.
v_{esc} = (2 * G * M / r)^{1/2}
where G is called the Gravitational constant. The notation
(2 * G * M / r)^{1/2}
means (2 * G * M / r) to the one-half power, which is equal to the square root of (2 * G * M / r).
You will calculate the escape velocity for a number of bodies using the MKS system where the units for distance are meters, the units for mass are kilograms, and the units for time are seconds. In this system, the gravitational constant has the value:
G = 6.67 * 10^{-11} Newton-meter^{2}/kilogram^{2}.
As an example, the mass M of the Earth is 5.98 * 10^{24} kilograms. The radius r of the Earth is 6378 kilometers, which is equal to 6.378 * 10^{6} meters. The escape velocity at the surface of the Earth can therefore be calculated by:
v_{esc} | = | (2 * G * M / r)^{1/2} |
= | ( 2 * (6.67 * 10^{-11}) * (5.98 * 10^{24}) / (6.378 * 10^{6}) ) ^{1/2} | |
= | 1.12 * 10^{4} meters/second | |
= | 11.2 kilometers/second |
So, as with surface gravity, a simple Physics equation can be used to calculate the escape velocity for a body (in this case the Earth) if you know the mass of the body and its radius! The assumption in using this formula is that the body is spherical, but this is a pretty good assumption. If the radius of a body at its equator and pole are very different, then the escape velocity is different at those places and should be calculated separately.
The escape velocity for the Earth is therefore 11.2 kilometers per second. This is the velocity that an object (or gas molecule!) needs at the surface of the Earth to be able to overcome the gravitational attraction of the Earth and escape to space.
A table of masses and radii is given below for many bodies in the Solar System. Make sure to convert the radii from kilometers to meters when making the calculation, and make sure that you can calculate the escape velocity of the Earth correctly. Then, calculate the escape velocity at each of the other bodies.
Body | Mass (kg) | Radius (km) |
---|---|---|
Earth | 5.98 * 10^{24} | 6378 |
Mercury | 3.30 * 10^{23} | 2439 |
Venus | 4.87 * 10^{24} | 6051 |
Mars | 6.42 * 10^{23} | 3393 |
Jupiter | 1.90 * 10^{27} | 71492 |
Saturn | 5.69 * 10^{26} | 60268 |
Uranus | 8.68 * 10^{25} | 25559 |
Neptune | 1.02 * 10^{26} | 24764 |
Pluto | 1.29 * 10^{22} | 1150 |
Moon | 7.35 * 10^{22} | 1738 |
Ganymede | 1.48 * 10^{23} | 2631 |
Titan | 1.35 * 10^{23} | 2575 |
Sun | 1.99 * 10^{30} | 696000 |