Acceleration and Motion Graphs

Acceleration measures a change in velocity

Two cars can have the same velocity at one instant but behave differently immediately afterwards. One may maintain its velocity while the other speeds up. Acceleration describes this difference: it is the rate at which velocity changes with time. A change in velocity can involve speed, direction or both. In this chapter the motion lies along one fixed line, so direction is represented by a sign.

The phrase “per second squared” does not mean that an object travels a square of time. Velocity already measures position change per second; acceleration measures how much that velocity changes per second. For example, velocities −6, −4 and −2 m/s at successive one-second instants show a constant +2 m/s² acceleration, although the object is initially moving in the negative direction.

Negative acceleration is not always slowing down

Away from v = 0, matching signs mean speed increases; opposite signs mean speed decreases. The sign of acceleration alone cannot answer that question. At v = 0, examine the motion on either side: with nonzero acceleration, an object may be turning around. “Retardation” or “deceleration” describes slowing down; when a numerical deceleration magnitude is quoted, it is normally positive even though the signed acceleration depends on the chosen axis.

Reading all three graphs together

On a position-time graph, slope gives velocity. Curvature describes how that slope changes: concave upward means increasing velocity and positive acceleration; concave downward means decreasing velocity and negative acceleration. Do not confuse “curve rises” with “acceleration is positive”. A rising curve that gradually flattens has positive velocity but negative acceleration.

On a velocity-time graph, height gives velocity and slope gives acceleration. A horizontal line above zero is constant positive velocity, not rest. A sloping straight line means constant acceleration. Where the line crosses zero velocity, the object reverses if the sign changes. On an acceleration-time graph, height gives acceleration, and signed area gives change in velocity.

You can understand the area rule without advanced calculus. Divide time into tiny intervals. In each interval, displacement is approximately velocity multiplied by the duration: a thin rectangle on the graph. Adding the rectangles and making them narrower approaches the exact signed area. Areas below the time axis are negative because the velocity there is negative.

Displacement and distance from the same graph

What zero and constant quantities really imply

An object at rest throughout an interval has zero velocity and zero acceleration throughout that interval. Momentary rest does not imply zero acceleration. Similarly, zero average acceleration only means equal endpoint velocities; acceleration could have varied in between. Constant nonzero speed along one line with smooth motion implies constant direction and zero acceleration. A sudden idealised reversal would require a special collision or discontinuity; do not apply a differentiable-motion statement at that instant.