Q: Kinematics: Understand, use and interpret graphs in kinematics for motion in a straight line: displacement against time and interpretation of gradient; velocity against time and interpretation of gradient and area under the graph.

The idea

Kinematics graphs describe motion along a straight line. On a displacement–time graph, the gradient at a point gives instantaneous velocity; a constant gradient means constant velocity, while a curve shows changing velocity. On a velocity–time graph, the gradient gives acceleration, and the area between the graph and the time axis gives displacement. Areas below the time axis count as negative displacement. For example, if v = 6 − 2t for 0 ≤ t ≤ 5, the gradient is −2 m s⁻², and the area from t = 0 to 3 is a triangle of area 9 m, while from t = 3 to 5 the area is −4 m, giving displacement 5 m. You must interpret signs, distinguish displacement from distance, and use gradients and areas to solve problems.

What earns the marks

• State that the gradient of a displacement–time graph gives velocity, and calculate it using a chord or tangent as appropriate.
• State that the gradient of a velocity–time graph gives acceleration, and calculate it from the graph.
• State that the area under a velocity–time graph gives displacement, and calculate areas using triangles, trapeziums or rectangles.
• Interpret negative gradients and areas below the time axis as motion in the opposite direction or negative displacement.
• Distinguish between displacement and total distance travelled when the velocity changes sign.
• Use graph features such as intercepts, turning points and asymptotes to describe motion qualitatively.

Common mistakes and how to put them right

• Confusing the roles of gradient and area.

  Put it right: Gradient of s–t gives velocity; area under v–t gives displacement.

• Ignoring the sign of areas below the time axis, treating all area as positive distance.

  Put it right: Areas below the axis contribute negative displacement; total distance is the sum of absolute areas.

• Assuming a straight line on a displacement–time graph means the object is stationary.

  Put it right: A straight line with non-zero gradient means constant velocity; only a horizontal line means stationary.

Exam tips

• Label axes with quantities and units before reading values, and show the triangle or trapezium you use for gradient or area.
• When asked for distance travelled, split the motion at points where v = 0 and add the magnitudes of each area.
• Check whether the question asks for a value at a specific time or over an interval; use a tangent for instantaneous rates and a chord for average rates.