Velocity-Time Graphs

A velocity–time graph carries two quantities at once, which is what makes it the most useful graph in mechanics.

The gradient is the acceleration — because acceleration is rate of change of velocity. A straight line means constant acceleration; a horizontal line means constant velocity and therefore zero acceleration.

The area under the line is the displacement — check by units: m s⁻¹ multiplied by s gives m. Split the area into triangles and rectangles and add them.

Example: a car accelerates uniformly from rest to 20 m s⁻¹ in 4.0 s, then holds that speed for 6.0 s. The triangle gives ½ × 4.0 × 20 = 40 m, the rectangle gives 6.0 × 20 = 120 m, so the total displacement is 160 m.

Area below the axis counts as negative — a negative velocity means motion in the opposite direction, so that area subtracts from the displacement. An object that goes out and comes back has equal areas above and below, and a total displacement of zero, even though it has certainly travelled.

Distance versus displacement from the graph — for distance, add the magnitudes of all the areas, ignoring sign. For displacement, subtract the areas below the axis. A question asking for total distance travelled is testing exactly this.

A curved velocity–time graph — the acceleration is changing. Find the instantaneous acceleration by taking the gradient of a tangent, and estimate the area by counting squares or by splitting the shape into strips.

Reading the shape — a line sloping down towards the axis means decelerating; a line crossing the axis means the object has reversed direction. It is worth saying explicitly that a negative gradient does not always mean slowing down: if the velocity is already negative, a negative gradient means speeding up in the negative direction.