Escape Velocity

Lesson 7 of 10 · 7 min · Free, no account needed

What you will learn

A planet sits in a gravity well: to leave for good you need enough energy to climb out of it.
Escape speed v = √(2GM / R) is √2 times the circular speed at the surface: about 11.2 km/s from Earth.
Small worlds have shallow wells: about 2.4 km/s on the Moon, and a jump is enough on an asteroid a few km across.

On a small enough asteroid, you could jump off into space. How small?
Fire a ball straight up from Earth. How fast before it never comes back?
Gravity makes a well around a planet. Give the ball just enough energy to climb out.
Escape velocity
The depth of the well per kilogram is GM / R. Kinetic energy ½ v² must match it: v_esc = √(2GM / R).
Circular speed at the same place is √(GM / R), so escape speed = √2 × circular speed. For Earth: 7.9 km/s and 11.2 km/s.
Simplified model: no air, and we ignore the pull of the Sun and other planets.
Put each world into the well that fits it.
v_esc = √(2GM / R), so the depth of the well goes with M / R. Jupiter has 318 Earth masses in 11 Earth radii: about 28 times Earth's M / R.
Game: Asteroid jumper. Hop from A to B to C.
These rocks are 3 to 5 km across. Too fast and you leave for good, too hard and you bounce.
AI: training a model can get stuck in a shallow pit of its error. Which kick gets it out?
To leave a pit the ball needs energy at least equal to the height of the bump. The small kick gives less, so it rolls back.
Circular speed at Earth's surface is 7.9 km/s. What is the escape speed?
v_esc = √(2GM / R) and v_circ = √(GM / R), so v_esc = √2 × 7.9 km/s, about 11.2 km/s.
Your route across the asteroids. Press Play.

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