Escape speed

How fast to never come back?

Narrated video. Transcript:

Throw a ball straight up. It slows down, stops and falls back. Throw it faster and it goes higher before it returns.

Is there a speed so large that the ball never comes back? Let us ignore air resistance and use energy to find out.

On the surface, the ball has kinetic energy half m v squared, and potential energy minus G M m over R. Watch the two energy bars side by side.

As it rises, kinetic energy turns into potential energy. To just reach infinity, it can arrive there with zero speed. At infinity, the potential energy is also zero.

So the total energy at the start must be at least zero: half m v squared minus G M m over R is greater than or equal to zero.

The smallest such speed is the escape speed: v e equals the square root of 2 G M over R. Since G M equals g R squared, this is also the square root of 2 g R.

Put in g = 9.8 and R = 6.4 × 10⁶ metres. The escape speed from the Earth is about 11.2 kilometres per second, roughly 40,000 kilometres per hour.

Notice that m cancelled out. A pebble and a rocket need the same escape speed. And if we ignore the air, the direction of launch does not matter either.

Why the Moon has no air

Narrated video. Transcript:

Escape speed depends on the mass and radius of the body: the square root of 2 G M over R. A small, light body has a small escape speed.

The Moon’s mass is about one eighty-first of the Earth’s, and its radius is about a quarter. Its escape speed works out to about 2.3 kilometres per second, nearly five times smaller than the Earth’s.

Now zoom in on a gas. Its molecules zip around in all directions at different speeds. A few of the fastest ones move at a good fraction of a kilometre per second or more.

On the Moon, during the long hot lunar day, molecules that reach the escape speed fly off into space and never come back. Watch them leak away one by one.

Over millions of years, any gas the Moon had has escaped. That is why the Moon has no atmosphere, and why astronauts there needed spacesuits.

The Earth’s larger escape speed holds on to nitrogen and oxygen, so we keep the air we breathe.

Try it: launch from the Earth

Interactive simulation. Things to try:

• The launch is sideways from the surface (no air). Press Launch or Space.
• Below about 7.9 km/s it falls back. At 7.9 km/s it circles the Earth.
• Between 7.9 and 11.2 km/s it moves on longer and longer ellipses.
• At 11.2 km/s and above, the total energy is zero or positive and it escapes.

Quick check

A 1 kg ball and a 1000 kg rocket are launched from the Earth’s surface. Ignoring air resistance, how do their escape speeds compare?

1. The rocket needs a speed 1000 times larger
2. The rocket needs a speed √1000 times larger
3. Both need the same speed, about 11.2 km/s
4. The ball needs a larger speed because it is lighter

A different planet

A planet has twice the mass of the Earth and half its radius. If the escape speed from the Earth is 11.2 km/s, what is it from this planet?

1. 11.2 km/s
2. 15.8 km/s
3. 22.4 km/s
4. 44.8 km/s

Summary

• Condition to escape: ½mv² − GMm/R ≥ 0
• Escape speed: vₑ = √(2GM/R) = √(2gR)
• From the Earth: vₑ ≈ 11.2 km/s

• A body escapes if its total energy is zero or positive, so it can reach infinity.
• Escape speed does not depend on the mass of the body (and, ignoring air, not on the direction of launch).
• It depends only on the planet’s M and R. The Moon’s escape speed is only about 2.3 km/s.
• Gas molecules can reach the Moon’s low escape speed, so the Moon has no atmosphere.