Motion graphsEdexcel International A Level Physics: Revision notes

A displacement–time graph plots displacement $s$ against time $t$. Its gradient is velocity, $v = \frac{\Delta s}{\Delta t}$.

• A straight sloping line means constant velocity
• A horizontal line means the object is at rest
• A negative gradient means motion in the opposite direction
• A curve means changing velocity; the velocity at an instant is the gradient of the tangent to the curve at that point

Average velocity is total displacement divided by total time, so it is zero for a return journey.

A velocity–time graph plots $v$ against $t$.

• The gradient is acceleration, $a = \frac{\Delta v}{\Delta t}$
• The area between the line and the time axis is displacement
• A horizontal line is constant velocity (zero acceleration)
• A straight sloping line is uniform acceleration; a negative gradient is a negative acceleration
• Areas below the time axis are negative displacement

For a trapezium or triangle use the area formulae; for a curve, divide into strips and use the trapezium rule or count squares.

An acceleration–time graph plots $a$ against $t$. The area under it gives the change in velocity, $\Delta v$.

• Horizontal line: constant acceleration
• Line at zero: constant velocity
• Falling but positive line: velocity still increasing, but more slowly

The gradient has little use at this level. To find the velocity at a given time, add the area to the initial velocity.

Example: $3.0\ \text{m s}^{-2}$ for $4.0\ \text{s}$ gives $\Delta v = 12\ \text{m s}^{-1}$; then falling uniformly to zero over $6.0\ \text{s}$ adds $\frac{1}{2} \times 3.0 \times 6.0 = 9.0\ \text{m s}^{-1}$.

• Displacement–time: gradient is velocity
• Velocity–time: gradient is acceleration, area is displacement
• Acceleration–time: area is change in velocity

Use the units to check: m ÷ s = m s⁻¹ (gradient of s–t), m s⁻¹ ÷ s = m s⁻² (gradient of v–t), and m s⁻¹ × s = m (area of v–t).

When acceleration is not constant, the velocity–time graph is a curve. The acceleration at an instant is the gradient of the tangent at that time. The distance is found by counting squares or by the trapezium rule: area ≈ strip width × (half the sum of the first and last values + the sum of the others).

A falling object in a fluid shows terminal velocity: the resistive force rises as it speeds up until it equals the weight, the resultant force and so the acceleration are zero, and the velocity-time graph becomes horizontal.

Example: speeds of 0, 0.15, 0.26, 0.33, 0.37 and 0.39 m s⁻¹ at 0.20 s intervals give an area $0.20 \times [0.195 + 0.15 + 0.26 + 0.33 + 0.37] = 0.26\ \text{m}$.