Motion graphs

Displacement-time graph

The gradient of a displacement-time graph at a particular time gives the velocity of the object at that time.

\text{Gradient} = \frac{{change\,in\,displacement}}{{change\,{in} \,t}} = \frac{{\Delta s}}{{\Delta t}} = \text{velocity}

Velocity-time graph

The gradient of a velocity-time graph at a particular time gives the acceleration of the object at that time.

\text{Gradient} = \frac{{change\,in\,velocity}}{{change\,{in} \,t}} = \frac{{\Delta v}}{{\Delta t}} = \text{acceleration}

The area under a velocity-time graph gives the displacement.

The velocity-time and acceleration-time graphs for common motions are shown below.

For the constant positive velocity graphs and the constant positive acceleration graphs, the initial displacement is zero and the initial velocity is zero.

For the constant negative acceleration graphs the initial displacement is negative and the initial velocity is positive.

Velocity-time and acceleration-time graphs for constant positive velocity

Velocity-time and acceleration-time graphs for constant positive acceleration

Velocity-time and acceleration-time graphs for constant negative acceleration

Question

Which of the three motion graphs do you think is most useful?

The velocity time graphs give all three variables for motion–velocity which can be read off at any time. Area is displacement and gradient is acceleration.