### classical mechanics

**Space station catch**

**Last modified: 2014-10-21**

A space station in the shape of a rigid disk of radius $R$ is spinning in outer space. As viewed by an inertial observer, the disk is spinning counterclockwise with constant angular velocity $\omega$ about an axis perpendicular to its surface and passing through its center. Astronaut Josh's feet are strapped onto the disk on its very edge, and Sandra Bullock's feet are strapped onto the diametrically opposite point.

Josh needs to throw a wrench to Sandra. If he decides to throw the wrench with a speed $R\omega$ parallel to the plane of the disk as viewed from a frame rotating with the disk, at what angles $\theta$ can he throw the wrench so that Sandra will be able to catch it?

Assume that $\theta$ is defined so that $\theta=0$ corresponds to throwing the wrench directly at Sandra, while $\theta = \pi/2$ corresponds to throwing the wrench in the direction directly opposite the rotation of the disk. Approximate Josh, Sandra, and the wrench as point particles, and assume that after the wrench is released, it moves as a free particle.