## Thursday, April 5, 2012

### Spaceship speed

A spaceship has two tanks for two different kinds of fuel. Tank A will be exhausted after one hour, where tank B will be exhausted after $\sqrt{p}$ hours, where $p$ is a prime number. Each tank is being depleted at a constant rate, and when empty, immediately refilled to full from an unlimited reserve. Assume that the refilling time is negligible.

Suppose $V$ represents the spaceship speed at time zero, and $r$ represents the lesser of the ratio between fuels in tank A or B versus their respective capacity. The spaceship speed at any given time is $rV$.

If the tanks are both full at time zero, and the ship travels a really long distance, what is the average speed of the ship? Express in terms of $V$ and $p$.