CAIE A-Level · Mathematics 9709 · Kinematics of Motion in a Straight Line

Displacement–Time and Velocity–Time Graphs (9709 Mechanics 4.2)

9 min readSyllabus 4.2PreviewBy Uzair Khan

Syllabus objective

Sketch and interpret displacement–time graphs and velocity–time graphs, and in particular appreciate that the area under a velocity–time graph represents displacement, the gradient of a displacement–time graph represents velocity, and the gradient of a velocity–time graph represents acceleration.

Introduction

Graphical kinematics is one of the most reliable sources of marks in the 9709 Mechanics paper. Rather than computing algebraically, questions ask you to read, sketch, and interpret two types of motion graph. A firm understanding of what gradient and area mean on each graph type allows you to extract every kinematic quantity — displacement, velocity, and acceleration — directly from the picture.


Core Concept

Two graph types capture all one-dimensional motion:

Displacement–time (s–t) graph — the vertical axis shows displacement ss from a fixed origin; the horizontal axis shows time tt.

Velocity–time (v–t) graph — the vertical axis shows velocity vv; the horizontal axis shows time tt.

Three fundamental interpretations connect these graphs:

GraphWhat the gradient givesWhat the area gives
Displacement–time (ss vs tt)Velocity vv— (not directly used)
Velocity–time (vv vs tt)Acceleration aaDisplacement ss (signed)

These follow directly from the definitions v=dsdtv = \dfrac{ds}{dt} and a=dvdta = \dfrac{dv}{dt}, which you met in the prerequisite topic.

Displacement–Time Graphs — Key Features

  • A positive gradient means the object moves in the positive direction.
  • A negative gradient means the object moves in the negative direction.
  • A zero gradient (horizontal line) means the object is stationary.
  • A curved section means the velocity is changing (non-uniform motion).
  • The steeper the line, the greater the speed.

Velocity–Time Graphs — Key Features

  • A positive gradient means positive acceleration (speeding up in positive direction, or slowing down in negative direction).
  • A negative gradient means negative acceleration (deceleration when moving in the positive direction).
  • A zero gradient (horizontal line) means constant velocity.
  • Area above the tt-axis contributes positive displacement; area below contributes negative displacement.
  • Total distance = sum of the absolute values of all areas (regardless of sign).

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