M9.2MechanicsCore

Velocity-Time Graphs

A velocity-time graph carries two key pieces of information at once: the gradient is the acceleration, and the area under the line is the distance travelled. That makes it the most useful graph in kinematics.

25 min Video by Zeeshan Zamurred Constant Acceleration

Edexcel AS Level Maths: 9.2 Velocity-Time GraphsWatch the full walkthrough before the notes below.

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What you'll be able to do

Use gradient = acceleration
Use area under the graph = distance
Interpret a v–t graph
Calculate distance from a trapezium area

Gradient is acceleration

On a velocity-time graph, the $gradient$ is the $acceleration$ . A horizontal line means constant velocity (zero acceleration); a negative gradient means deceleration.

acceleration = gradient of the v - t graph

Slope = acceleration.

Area is distance

The $area$ between the line and the time axis equals the $distance$ travelled. For straight-line segments this is found by splitting the region into rectangles and triangles, or using the trapezium area.

distance = area under the v - t graph

Often a trapezium:

\frac{1}{2} (a + b) h

1Area of triangle:

\frac{1}{2} \times 10 \times 20

Answer

100

Tip — v–t graph: gradient = acceleration, AREA = distance. Don’t mix these up with the s–t graph.

Reading the motion

A rising line is acceleration, a falling line is deceleration, and a horizontal line is constant velocity. Where the line meets the time axis ( $v = 0$ ), the object is momentarily at rest.

Formula recap

a = gradient of v - t graph

Acceleration from slope.

distance = area under v - t graph

Distance from area.

trapezium = \frac{1}{2} (a + b) h

Common area formula.

Common mistakes to avoid

Reading the gradient of a v–t graph as the velocity.

On a v–t graph the gradient is the ACCELERATION; velocity is the height.

Forgetting that the area gives distance.

Area under the v–t graph = distance travelled.

Key takeaways

v–t graph: gradient = acceleration, area = distance.
Horizontal line = constant velocity; falling line = deceleration.
Find distance via rectangle/triangle/trapezium areas.

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