Velocity and acceleration in SHM

You will write the velocity and the acceleration of a body in SHM, and then check the velocity with a small step in time.

The idea. For $x = A\cos\omega t$:

• velocity $v = -A\omega\sin\omega t$;
• acceleration $a = -\omega^2 A\cos\omega t = -\omega^2 x$;
• speed at the position $x$: $|v| = \omega\sqrt{A^2 - x^2}$.

A velocity is also the change of position divided by a small time. So $v \approx \dfrac{x(t + h) - x(t - h)}{2h}$ for a small $h$.

Your tasks.

1. In velocity, return $v$ at the time t.
2. In acceleration, return $a$ at the time t.
3. In speed_at, return the speed at the position x.
4. In step_velocity, return the estimate of $v$ from the positions at t + h and t - h.