2009
Jumping off a small world
- Round
- Theory
- Points
- 10
- Level
- Bronze
- Taught in
- Section 3.10
Adapted from IOAA 2009 (Tehran), theory problem 3. We have restated it in our own words and kept the original data. The solution was last checked on 21 October 2026.
Estimate the radius of a planet from which a person could escape by jumping straight up. Assume the planet has the same density as Earth, 5514 kg/m³.
Find the take-off speed from the height of an ordinary jump on Earth, and write the escape speed in terms of the radius and the density.
- A person can jump about 0.5 m high on Earth, so their take-off speed is
- With
- Escape needs
AnswerThe planet's radius would have to be at most about 1.8 km. Escape speed grows in proportion to the radius at a given density, which is why small asteroids cannot hold anything loose on their surfaces.
This problem also appears among the olympiad problems of Section 3.10, The escape of atmospheres, which teaches the ideas it uses. Equation and figure numbers refer to the course.