Area Under a Speed-Time Graph

The area between the speed-time graph line and the time axis equals the distance travelled. This relationship connects graphical analysis to kinematics calculations.

Why does the area equal distance?

Speed multiplied by time gives distance ($s = vt$). On a speed-time graph, speed is on the vertical axis and time is on the horizontal axis. The product of these two axes (height times width) is the area, and it equals the distance.

How do you calculate the area?

For simple shapes:

• Rectangle: area $= v \times t$ (constant speed section)
• Triangle: area $= \frac{1}{2} \times \text{base} \times \text{height}$ (uniform acceleration from or to zero)
• Trapezium: area $= \frac{1}{2}(a + b) \times h$ (uniform acceleration between two non-zero speeds)

For curved graphs, count the squares under the line and multiply by the value of one square.

Worked example

A car accelerates uniformly from rest to 20 m/s in 10 s, then travels at constant speed for 15 s. Calculate the total distance. [3]

Acceleration phase (triangle): $s_1 = \frac{1}{2} \times 10 \times 20 = 100\ \text{m}$

Constant speed phase (rectangle): $s_2 = 20 \times 15 = 300\ \text{m}$

Total: $s = 100 + 300 = 400\ \text{m}$

Marking points: M1 for correct area of triangle. M1 for correct area of rectangle. A1 for 400 m.

These marking points are for this original example and are not an official Cambridge mark scheme.

Common errors and how to correct them

• Forgetting the ½ in the triangle formula. Using base × height without the ½ doubles the answer. If the shape has a sloping side, it is a triangle or trapezium, not a rectangle.
• Calculating area above the line instead of below. The distance is the area between the graph line and the time axis, not the area above the line.
• Miscounting squares on a curved graph. Count squares that are more than half filled as full squares. Squares less than half filled are ignored. State the scale of each square clearly.

How to apply this in an exam

Split the area into recognisable shapes (rectangles and triangles). Calculate each area separately, show the working for each, then add them. Label which section of the journey each area represents.