5.6.12Forces

Area Under Velocity-Time Graphs

5 min read 5 parts Lesson 48 of 67 Unit quiz

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Section 1 of 5Reading a Velocity-Time Graph

A velocity-time graph plots velocity on the vertical axis against time on the horizontal axis, so both its gradient and its area carry physical meaning.

Gradient

Gradient The gradient of a velocity-time graph gives the acceleration, measured in metres per second squared.

Area

Area The area between the line and the time axis gives the distance travelled during that time interval.

Horizontal line

Horizontal line A horizontal line shows constant velocity, so the area under it is a rectangle of width time and height velocity.

Below the axis

Below the axis An area below the time axis represents motion in the opposite direction, so it counts as negative displacement.

Section 2 of 5Finding the Area

Complicated graphs are split into simple shapes, and the areas are added together.

Split the area under the graph into simple shapes, such as rectangles and triangles.

Calculate the area of a rectangle as width × height, which is time × velocity.

Calculate the area of a triangle as ½ × base × height, where the base is a time interval and the height is a change in velocity.

Add the areas of all the shapes together to find the total distance travelled.

Check the units: velocity in m/s multiplied by time in s gives a distance in metres.

Section of graph	Shape	Area formula	Physical meaning
Constant velocity	Rectangle	v × t	Distance covered at a steady speed
Uniform acceleration from rest	Triangle	½ × t × v	Distance covered while speeding up
Stationary	No height	0	No distance travelled

Section 3 of 5Connecting Gradient and Area

Connecting Gradient and Area

Two quantities

Two quantities The gradient gives acceleration, while the area gives distance, so the two calculations answer different questions.

Uniform acceleration

Uniform acceleration A straight sloping line means constant acceleration, so the area beneath it is a triangle or a trapezium.

Negative areas

Negative areas Area below the time axis counts as negative displacement, so total displacement can be smaller than the total distance.

Section 4 of 5Worked Example

Question

A car accelerates uniformly from 0 to 20 m/s in 10 s, then travels at 20 m/s for a further 30 s.
Calculate the total distance travelled.

Stage 1, a triangle: area = ½ × 10 × 20 = 100 m
Stage 2, a rectangle: area = 20 × 30 = 600 m
Total distance = 100 + 600 = 700 m

Answer The car travels a total distance of 700 m.

Section 5 of 5Exam Checklist & Tips

5.6.12 – Key Points for the Exam

Velocity-time graph A graph of velocity against time whose gradient gives acceleration and whose area gives distance.
Acceleration The rate of change of velocity, found from the gradient of a velocity-time graph.
Displacement The distance travelled in a stated direction, which can be negative for motion in the opposite direction.
Trapezium A four-sided shape with one pair of parallel sides, whose area is half the sum of the parallel sides multiplied by the height.
Uniform acceleration Acceleration that stays constant, shown as a straight sloping line on a velocity-time graph.

Common Mistakes

Using the gradient of the line to find distance; the area under the line gives the distance travelled.
Forgetting to halve the rectangle area when the shape is a triangle, as when acceleration starts from rest.
Reading the height of the line at one point as the total distance covered.
Forgetting to convert minutes to seconds before using the area, which gives a wrong answer in metres.

Physics 0625 & 9702 SUVAT Mastery Graph Interpretation

Constant Acceleration Equations (SUVAT) & Slopes

v = u + at | s = ut + ½at² | v² = u² + 2as | s = ½(u + v)t

Displacement-Time Graph: Gradient = Velocity (curve = accelerating).
Velocity-Time Graph: Gradient = Acceleration; Area under curve = Distance travelled / Displacement.
Free Fall in vacuum: a = g ≈ 9.81 m/s² downwards (air resistance negligible).