Position, Velocity, and Acceleration Together

Think of a mass on a spring sliding through equilibrium. At that instant the spring is relaxed, so the restoring force — and therefore acceleration — is zero. But the mass has been speeding up the whole way there, so this is exactly where its speed peaks. Push on to the endpoint of the swing: the mass stops for an instant, velocity zero, but the spring is stretched or compressed as far as it goes, so the restoring force and acceleration are at their max. This is the core pattern of 7.3: acceleration is proportional to displacement (a = -w² x), so whatever position is doing, acceleration mirrors it, while velocity does the opposite.

Graphed against time, position, velocity, and acceleration are all the same sine-shaped curve, just shifted. If position is x = A cos(wt), velocity is v = -Aw sin(wt), and acceleration is a = -Aw² cos(wt). Each one is a quarter cycle out of step with the next: velocity leads position by a quarter cycle, acceleration leads velocity by another quarter cycle, and acceleration ends up exactly opposite position, pointing back toward equilibrium at all times. Stack the three graphs and you'll see velocity cross zero where position peaks, and acceleration peak where position peaks but with the opposite sign.

This ties directly to the energy picture of oscillation: kinetic energy is greatest at equilibrium (where speed peaks) and potential energy is greatest at the endpoints (where acceleration peaks and the restoring force is strongest). Seeing position, velocity, and acceleration together is what makes that energy trade-off make sense.