Graphs of motion

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

$Gradient = {{change\;in\;displacement} \over {change\;in\;t}} = {{\Delta s} \over {\Delta t}} = velocity$

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

$Gradient = {{change\;in\;velocity} \over {change\;in\;t}} = {{\Delta v} \over {\Delta t}} = acceleration$

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.

Speed time graph showing constant positive velocity - has a positive and constant gradient, where the speed is directly proportional to the time. The velocity time graph (showing accelearation) for constant positive velocity shows a straight horizontal line - this shows acceleration of zero. The acceleration time graph is a graph of acceleration over time, but because the acceleration is zero, no line appears on the graph.

Speed time graph for constant positive acceleration: gradient (velocity) is positive and increasing by the same amount each time. Velocity time graph for constant positive acceleartion: gradient (acceleration) is positive and constant. Acceleration time graph for constant positive acceleration: straight horizontal line

Graphs to show constant negative acceleration. The gradient of speed time graph (which shows velocity) is positive and decreasing by the same amount each time, resulting in a increasing curve. The gradient for the velocity time graph (which shows acceleration) is negative and constant. The acceleration time graph shows a straight horizontal line which lies below zero.

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Detail from the Physics game Cannon.

Take aim and fire with the forces game Cannon.

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You can practise calculating equations of motion with our game, Cannon. Using velocity, gravity and other variables, try to predict where the cannonball will land.

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