​Displacement, velocity and acceleration for simple harmonic motion

​​In a nutshell

Any object undergoing simple harmonic motion will undergo the same displacement, velocity and acceleration relationships. Displacement is a maximum at the amplitude of the motion, velocity is a maximum through the midpoint and acceleration is a maximum when displacement is a minimum. 

Displacement time graph

Consider an object undergoing simple harmonic motion. If the object starts at a maximum displacement of $$A$$, the graph of displacement against time would look like this: 

The object would start at the maximum possible value of positive amplitude $$A$$. It would then pass through the midpoint until it reaches a minimum, or maximum negative, amplitude $$-A$$. 

The object would then move back towards the midpoint, eventually reaching its starting point of maximum positive amplitude $$A$$. This would then be one complete cycle and would take $$T \, s$$. The object would then repeat the same cycle.

Velocity time graph

The velocity time graph for the same object above would look very similar. First consider when the object would have its maximal and minimal values of velocity.

At the extremes of the displacements, the object completely changes direction, which means for a brief moment in time, the velocity of the object must be zero as it passes from positive velocity to negative velocity, or vice versa. 

The velocity will take a minimum at the displacements maxima and minima. The objects acceleration is always directed towards the midpoint, which means as soon as it passes through the midpoint, the acceleration switches, causing the object to slow down. 

This means that the object will have a maximum velocity through the midpoint. The maximum and minimum values for velocity are $$v_{max}=\omega A$$​ and $$v_{max}=-\omega A$$ respectively. 

Acceleration time graph

The acceleration time graph is going to be the opposite of the displacement time graph as the condition for simple harmonic motion is that acceleration is directly proportional to the displacement and is always towards the midpoint. 

At the extremes of the displacement, the acceleration will be a maximum and minimum. Taking a minimal value of $$-\omega^2A$$ when displacement is at a maximum value and inversely, taking $$\omega ^2A$$ at a minimum displacement.

Simple harmonic motion graphs

Putting all the graphs together, it is possible to see how the displacement, velocity and acceleration are related to one another for an object undergoing simple harmonic motion.