Motion graphs: x–t, v–t, a–t
Three linked graphs describe the same motion. Read the slope, read the area, and tell uniform from non-uniform.
One motion, three graphs: where it is, how fast it is going, how quickly that speed is changing. Each graph hides the other two inside its slope and its area.
On a position–time graph, a steeper straight line means…
The area under a velocity–time graph gives…
A flat, horizontal line on a v–t graph means…
The v–t graph of an object rises steadily from 5 m s⁻¹ to 25 m s⁻¹ in 4 s. Its acceleration, the gradient, is…
The area under an a–t graph gives…
A cyclist starts at 2 m s⁻¹ and speeds up uniformly to 8 m s⁻¹ in 6 s. Sketch the v–t graph in words, find the acceleration from its gradient, and find the distance from its area.
Displacement is…
For uniform motion, Y = …
Key points
- • x–t slope = velocity; v–t slope = acceleration; v–t area = displacement; a–t area = change in velocity.
- • Uniform motion: x–t is a straight line, v–t is flat, a = 0.
- • Non-uniform motion: x–t curves; the tangent at a point gives the instantaneous velocity; x = ut + ½at².
Formulas
- v = Δx / t
- x_f = x_i + ut + ½at²
- area under v–t = displacement
Key terms
- Gradient :
- rise / run, the slope of a graph
- Tangent :
- a straight line touching a curve at one point; its slope is the instantaneous gradient
Common errors
- • Reading the height of a v–t graph as the distance instead of the AREA.
- • Confusing "flat v–t line" with "at rest".
- • Measuring a gradient with a chord instead of a tangent.