Significance of areas of velocity–time and acceleration– time graphs and…

The idea

Graphs link motion variables through gradients and areas. On a displacement–time graph, the gradient gives velocity; a straight line means uniform velocity, while a curve means changing velocity. A parabolic curve indicates uniform acceleration. On a velocity–time graph, the gradient gives acceleration and the area between the line and the time axis gives displacement. Uniform acceleration yields a straight line, whereas non-uniform acceleration yields a curve. On an acceleration–time graph, the area gives the change in velocity. For a bouncing ball, each bounce shows a steep velocity change at impact, then a straight-line segment with a constant gradient as it moves under gravity, with decreasing peak speeds and shorter times between bounces.

What earns the marks

• Gradient of a displacement–time graph equals velocity (m s⁻¹); a straight line indicates uniform velocity, while a curve indicates changing velocity (a parabolic curve represents uniform acceleration).
• Gradient of a velocity–time graph equals acceleration; a straight line indicates uniform acceleration, while a curve indicates non-uniform acceleration.
• Area under a velocity–time graph equals displacement; areas below the time axis represent negative displacement, which must be subtracted from positive areas to find net displacement.
• Area under an acceleration–time graph equals change in velocity.
• For a bouncing ball, the velocity–time graph shows sharp vertical changes at each impact and parallel straight-line segments between impacts, with decreasing peak speeds as energy is dissipated.

Common mistakes and how to put them right

• Assuming any curved displacement–time graph means non-uniform acceleration.

  Put it right: A curve simply means velocity is changing. If the curve is a perfect parabola, the acceleration is uniform.

• Confusing gradient and area.

  Put it right: Area under a displacement–time graph has no direct physical meaning in this context; use the gradient to find velocity.

• Ignoring signs when finding displacement from a velocity–time graph.

  Put it right: Areas below the time axis are negative, so subtract them from areas above the axis to find net displacement.

• Using a chord instead of a tangent to find a gradient at a point on a curve.

  Put it right: Draw a tangent at the point and calculate its gradient to obtain the instantaneous value.

Exam tips

• For curved graphs, draw a tangent at the required point and clearly show the coordinates you use in the gradient calculation to find instantaneous velocity or acceleration.
• When calculating areas, split the region under the graph into simple geometric shapes like triangles, rectangles or trapezia, and state each area with its sign before combining them.