2009

Escaping from the centre of a cloud

Round: Theory
Points: 10
Level: Silver
Taught in: Section S4

Adapted from IOAA 2009 (Tehran), theory problem 6. We have restated it in our own words and kept the original data. The solution was last checked on 16 October 2026.

A cloud of gas from which a star will form has a uniform density, a mass $M$ and a radius $R$ . An object is launched from the centre of the cloud and moves through it without colliding with any of its particles. An object that fell freely from the surface of the cloud would reach the centre at a speed of $\sqrt{G M / R}$ . Find the speed at which the object must be launched from the centre to escape from the cloud altogether.

I solved it without the solution

AnswerThe object must be launched at $v = \sqrt{3 G M / R}$ , which is $\sqrt{3 / 2}$ , about 1.22, times the escape speed from the surface of the cloud.

This problem also appears among the olympiad problems of Section S4, Physics toolkit, which teaches the ideas it uses. Equation and figure numbers refer to the course.