How quickly does a swing die away?

You will write three functions for a damped oscillator.

The idea. With a damping force $-bv$, the amplitude after a time $t$ is $A_0e^{-bt/2m}$. Set it equal to a fraction of $A_0$ and take the natural log to find the time.

Your tasks.

1. In amplitude, return the amplitude after a time t, in the unit of A0.
2. In half_time, return the time in s for the amplitude to fall to half.
3. In time_to_fraction, return the time in s for the amplitude to fall to the fraction frac of its start.