Motion is the first place where the difference between a scalar and a vector earns its keep. Distance is not displacement, speed is not velocity, and a graph quietly hides two more facts in plain sight: its gradient and the area beneath it each mean something physical.
Displacement, velocity and acceleration are vectors; distance and speed are their scalar cousins. On a displacement-time graph the gradient is the velocity. On a velocity-time graph the gradient is the acceleration and the area beneath the line is the displacement. Read the graph correctly and the numbers fall out.
Set an initial velocity and an acceleration, then run the trolley. As the velocity-time line builds, its gradient stays equal to the acceleration and the shaded region beneath it grows into the displacement. Predict what that shaded area represents before you check.
Two graphs, four quantities. Keep gradient and area in the right column and most kinematics questions become routine.
| Graph | Gradient gives | Area beneath gives |
|---|---|---|
| displacement against time | velocity | (no standard meaning) |
| velocity against time | acceleration | displacement |
Definitions to keep crisp: velocity is the rate of change of displacement, v = Δs / Δt; acceleration is the rate of change of velocity, a = Δv / Δt. Both are vectors, so a sign tells you direction.
Four quick checks on scalars and vectors of motion and on reading graphs. Each correct answer earns XP and lights this skill on your star map.
A runner completes exactly one lap of a 400 m circular track and stops where they began. Their distance and displacement are:
On a displacement-time graph, the gradient of the line at a point gives the:
In the simulator, the shaded region under the velocity-time line represents the:
Acceleration is best defined as the:
Match the operation to the right graph. The area that gives displacement is the area under the velocity-time graph, not under the displacement-time graph, which has no standard meaning. Likewise the gradient of a displacement-time graph is velocity, while the gradient of a velocity-time graph is acceleration. A flat (horizontal) line on a displacement-time graph means the object is stationary, not moving at constant speed.
Unlocks once the four checks above are done. Worth more XP, written to AS Paper 1 and 2 standard.
A car moves at a constant 15 m s⁻¹ for 8.0 s. Using the area under its velocity-time graph, the distance travelled is:
A displacement-time graph curves upward with an increasing gradient. This tells you the object is:
A velocity-time graph is a straight line falling from 20 m s⁻¹ to 0 over 5.0 s. The acceleration is:
A cyclist rides 300 m east in 20 s, then 300 m back west in 30 s, returning to the start. Over the whole trip the average velocity and average speed are:
This skill is now lit gold on your star map. Keep the chain going.