Displacement / time graph

The graph below plots of displacement against time for a mass on a spring ruler oscillating in simple harmonic motion. It is sinusoidal. The rule in all the graphs on this page is that up is positive.

At any point the gradient of the graph is ds/dt where s is displacement and t is time which is equal to velocity. The yellow arrows indicate maximum displacement at the extremes of oscillation. At these point the mass stops for an instant, the velocity is zero.

Displacement time graph

Velocity / time graph

The velocity is always zero at each end of the oscillation where the displacement is a maximum. It is a maximum when the displacement is zero. The two sine waves are one quarter of a wave or 90 degrees or Π/2 radians out of phase.

The gradient of this graph at any point is dv/dt, which is the rate of change of velocity, the definition of acceleration.

Velocity / time graph compared to displacement / time graph

Acceleration / time graph

Acceleration time graph compared to displacement

Because the force always acts towards the centre and the displacement is measured from the centre outwards, the sine graph for acceleration is the reverse of that for displacement. The two graphs are half a wave or 180 degrees or Π radians out of phase.

Graph summary

Examination questions often ask you to compare these three graphs and the phase differences.

Summary of displacement, velocity and acceleration graphs