This activity will illustrate how to calculate the escape velocity of planets, satellites and the Sun.
Time requirement: 50 minutes as an activity for older or mathematically advanced students.

Materials

• Calculator (with square root function)

Procedures

You have learned that the escape velocity (v_esc) of a body depends on the mass (M) and the radius (r) of the given body. The formula which relates these quantities is:

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-meter²/kilogram².

As an example, the mass M of the Earth is 5.98 * 10²⁴ kilograms. The radius r of the Earth is 6378 kilometers, which is equal to 6.378 * 10⁶ meters. The escape velocity at the surface of the Earth can therefore be calculated by:

vesc	=	(2 * G * M / r)1/2
	=	( 2 * (6.67 * 10-11) * (5.98 * 1024) / (6.378 * 106) ) 1/2
	=	1.12 * 104 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 * 1024	6378
Mercury	3.30 * 1023	2439
Venus	4.87 * 1024	6051
Mars	6.42 * 1023	3393
Jupiter	1.90 * 1027	71492
Saturn	5.69 * 1026	60268
Uranus	8.68 * 1025	25559
Neptune	1.02 * 1026	24764
Pluto	1.29 * 1022	1150
Moon	7.35 * 1022	1738
Ganymede	1.48 * 1023	2631
Titan	1.35 * 1023	2575
Sun	1.99 * 1030	696000

Note that the Gas Giant planets (Jupiter, Saturn, Uranus and Neptune) do not have solid surfaces. The radii of these planets are specified at the point where the pressure in their atmospheres is approximately equal to that at the surface of the Earth. As one last exercise, convert the escape velocity of the Earth to kilometers per hour (or miles per hour) to get a good feeling for how much initial velocity an object must really have in order to escape the gravitational force of our planet!