    ESCAPE VELOCITY

You may occasionally her someone speak of "escaping a planet's gravity." This is really an inaccurate statement, since every object in the Universe pulls on every other object, however small that pull might be. The key to "escaping" something's gravity is to get far away from that something, so far away that its pull on you is just as ignorably small as all the other pulls that are tugging at you from all other objects. Remember that gravity gets weaker with distance!

To get to this distance, you must have a certain amount of kinetic energy (KE), the energy of motion. The bare minimum of kinetic energy that you must have is equal to the gravitational potential energy (PE) between you and the object (planet, moon, star, black hole, whatever) you are trying to get away from. If your KE is less than the object's PE, then you will gradually slow down, stop, and fall back down. Since kinetic energy depends strongly on the speed at which you are traveling, the minimum kinetic energy translates to a minimum velocity, called the escape velocity.

The formula for kinetic energy is

KE = (1/2) m v2

where m is your mass, or the mass of the thing you're trying to "set free," and v is that thing's velocity (NOT volume!). We set this kinetic energy equal to the gravitational potential energy, given by

PE = G m M / R

where m is once again the mass of the thing that is trying to escape, M the mass of the planet/star/whatever that the thing is escaping from, and R is the size (radius) of the planet/star/whatever that the thing is escaping from. G is Newton's Universal Constant of Gravitation. Setting KE = PE, we get

(1/2) m v2 = G m M / R

Notice that the mass of the thing that is escaping, m, drops out. It doesn't matter if you're launching a baseball or a Mack Truck or a space shuttle into space, they all have to achieve the same velocity! We rearrange the equation to get

v2 = 2 G M / R

Take the square root of both sides:

vesc = Square root (2 G M / R)

Plug in the relevant numbers for the Earth, and you should get a value for vesc of around 11 kilometers/second, or about 7 miles/second. This is how fast an object must be moving to "escape" the Earth's gravity, and "go up without coming down." Bear in mind that it still takes a lot more energy/force to get a high-mass object accelerated to the escape velocity, but the value of that velocity is the same for anything regardless of mass.

See that our beautiful equation above is marred by the presence of that nasty letter G. We can get around having to know the value of G by comparing the escape velocity of an unknown object to that of an object we know, such as the Sun or the Earth:

vesc / vEarth = Square root (2 G Mobj / Robj) / Square root (2 G MEarth / REarth)

vesc / vEarth = Square root (2 G Mobj REarth / 2 G MEarth Robj)

vesc / vEarth = Square root (Mobj REarth / MEarth Robj)

Both 2 and G drop out, and we are left with a much simpler equation! Units for mass and radius do not matter, so long as both mass units agree and both radius units agree.

Sample Calculations

To see some sample calculations with escape velocity, click on the examples below Updated 8/27/99
By James E. Heath

Copyright Ó 1999 Austin Community College