Problem

A particle moving back and forth on the x axis has its position given as a function of time t as follows:

Where is the particle when t = 0 and how long does it take to return to this position?

Solution

When t = 0, we have

The particle will return to this position whenever

The cosine function has the value 0 at p/2, 3p/2, 5p/2, and so on. We can see that for

to be one of the values in this series, pt must be a multiple of p; that is, t must be an integer. Therefore the particle returns to its starting position (when t = 0) at times t = 1, 2, 3, and so on.

We can see this when we graph the position function x(t).