**Target Age Group**

Middle School

**Ojbective**

Students will understand escape velocity, or the speed needed to escape the gravitational field of a body.

**You will need**

computer with internet service, pencil, paper

**Background**

Any body that has mass has an associated gravitational field, or “pull”. The more massive an object is, the greater its gravitational pull. This concept can be generalized for any object of mass. This force is proportional to the product of the two masses and inversely proportional to the square of the distance between the two masses:

where F = the magnitude of the gravitational force between the two masses, G is the gravitational constant (6.67 x 10-11 Nm2kg-2), m1 and m2 are the masses of the first object and second object, respectively, and r is the distance between the two objects. From this equation, you can see that if you were orbiting Earth at some distance r, and if m1 was your mass and m2 was the mass of the Earth, F would be very large. However, if you were orbiting Jupiter at the same distance r, and m2 was the mass of Jupiter, F would be much larger. Therefore, the more massive an object is, the greater its gravitational force.

In order to escape the gravitational force of a massive body, like a planet or a moon, an object on that body has to travel at a particular minimum speed, which we call the **escape velocity**. Since larger, more massive bodies exert a stronger gravitational force on objects, anything that wants to escape, or “break free” from that force has to travel at

a very high speed. Likewise, anything trying to escape the gravitational force of a less massive object can travel at a lower speed. Anything traveling at a speed less than the escape velocity will either fall back to or stay in orbit around the source.

**Equation and Finding Earth’s Escape Velocity:**

The escape velocity from a body is given by the following equation:

where ve is the escape velocity, G is the gravitational constant, M is the mass of the body being escaped from, and r is

the distance between the center of the body and the point at which the escape velocity is being calculated.

Let’s use this equation to derive the escape velocity of the Earth. Given that:

• G = 6.67 x 10-11 Nm2kg-2

(recall that 1N = 1kgm/s2)

• M = 5.9736 x 1024 kg

• r = mean radius of the Earth (if we want to escape from Earth’s surface)

= 6372.797 km

= 6,371,797 m

We will now plug these numbers in to find out ve :

Therefore, the escape velocity for the Earth is about

11.2 km/s, or roughly 7 mi/s.

So, the escape velocity for the Moon is about 2.38 km/s, or about 1.5 mi/s.

Starting from the calculations above, you can build a spreadsheet to calculate escape velocities for all the different bodies in the Solar System.

About Gravity Wells: For more about Gravity Wells, see the Gravity Well Animation and Earth’s Offshore Island: The Moon at the Google Lunar X PRIZE site for views of the relative ease of escaping from the Moon compared with Earth launching.

Now that you know the escape velocities of the Earth and the Moon, it’s easy to compare the relative energy required for Earth vs. Lunar escape. Energy equals the square of the velocity. So let’s square both of these and look at the ratio.

In other words, it takes 22 times more energy to launch material into free space from the Earth than it does from the Moon. So we say that the Earth’s gravity well is 22 times deeper than the “gravity dimple” of the Moon. In reality the benefits of lunar launching are even greater than this ratio since the Moon has no atmospheric drag.

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