Motion graphs represent how position, velocity, or acceleration varies with time. Reading them requires attention to the quantity on each axis, the sign of the ordinate, the gradient, and any physically meaningful signed area.
The three standard time graphs describe the same motion from different viewpoints:
- the gradient of a position-time graph gives velocity;
- the gradient of a velocity-time graph gives acceleration;
- the signed area between a velocity-time graph and the time axis gives displacement;
- the signed area between an acceleration-time graph and the time axis gives change in velocity.
A graph is not a picture of the path followed by the object. Its shape records how one physical quantity changes with time.
For a finite interval,
For a local value, use the gradient of the tangent to the relevant curve. Graph areas give
When translating between graphs, use the same routine each time:
- identify the axes, units, plotted ordinate, and any stated initial value;
- divide the motion into meaningful intervals and mark zero crossings or turning points;
- record the sign and whether the ordinate or gradient is constant, increasing, or decreasing;
- move to the next derivative using gradient, or reconstruct the previous quantity using signed area plus its initial value;
- check units, continuity, sign convention, and physical plausibility.
This interval-by-interval method is more reliable than matching a memorized graph shape.
Continuity provides an additional physical check. Position normally remains continuous because an object cannot change location instantaneously. Velocity also remains continuous unless the model explicitly includes an impulse-like idealisation. Acceleration may change abruptly when the force changes abruptly. At a position-time turning point, position remains continuous while its tangent gradient passes through zero and changes sign.