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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 Constant Acceleration
Edexcel AS Level Maths: 9.2 Velocity-Time GraphsCurated by Kepler, not produced by us. Watch it 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
1

Gradient is acceleration

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

acceleration=gradient of the v–t graph\text{acceleration} = \text{gradient of the } v\text{–}t \text{ graph}
Slope = acceleration.
2

Area is distance

The area\textbf{area} between the line and the time axis equals the distance\textbf{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\text{distance} = \text{area under the } v\text{–}t \text{ graph}
Often a trapezium: 12(a+b)h\tfrac{1}{2}(a + b)h.
Example 1A car accelerates from 0 to 20 m/s over 10 s (straight line). Find the distance.
1Area of triangle: 12×10×20\tfrac{1}{2} \times 10 \times 20.
Answer100100 m

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

3

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=0v = 0), the object is momentarily at rest.

Formula recap

a=gradient of v–t grapha = \text{gradient of } v\text{–}t \text{ graph}
Acceleration from slope.
distance=area under v–t graph\text{distance} = \text{area under } v\text{–}t \text{ graph}
Distance from area.
trapezium=12(a+b)h\text{trapezium} = \tfrac{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.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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Related lessons

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