Velocity and Acceleration in SHM

Start with position x(t) = A cos(omega t). Velocity is the rate of change of position, so v(t) = -A omega sin(omega t). Acceleration is the rate of change of velocity, so a(t) = -A omega² cos(omega t). Notice that a(t) = -omega² x(t): acceleration is always proportional to position and pointing back toward equilibrium. That single fact, a = -omega² x, is the defining equation of SHM, and everything about period and frequency comes from it.

These three quantities peak at different moments because sine and cosine are offset by a quarter cycle. Position is largest at the extremes, where the oscillator is momentarily stopped, so velocity there is zero. Velocity is largest at the equilibrium point, where x = 0, which is exactly where a = -omega² x also equals zero. Acceleration is largest at the extremes, right where velocity is zero. So speed is maximum when acceleration is zero and vice versa - they can never peak together, because one depends on sin(omega t) and the other on cos(omega t), and those never reach their maximum size at the same instant.

This is why maximum speed is v_max = A omega and maximum acceleration is a_max = A omega²: both come straight from the amplitude of the sine and cosine terms. These two numbers connect directly to the energy picture next - at v_max all the energy is kinetic, and at a_max (the extremes) all the energy is stored as potential energy in the spring or restoring force.