Distance–time and velocity–time graphs
These two graph types are the GCSE bread-and-butter of motion analysis.
Distance-time (d-t) graph
Plot distance (y) vs time (x).
- Gradient = speed. Steeper line = faster.
- Horizontal line = stationary.
- Curved line = changing speed.
- Curve sloping upward but flattening = slowing down.
- Curve becoming steeper = speeding up.
For a curved line, instantaneous speed is the gradient of the tangent at that point.
Velocity-time (v-t) graph
Plot velocity (y) vs time (x).
- Gradient = acceleration. Constant gradient = uniform acceleration.
- Horizontal line = constant velocity (acceleration = 0).
- Area under the line = distance travelled.
- Negative velocity = motion in reverse direction.
Calculating area for distance
- Triangle area: ½ × base × height.
- Rectangle: base × height.
- Composite shape: split into rectangles and triangles, sum the areas.
✦Worked example— Worked example — v-t graph
Velocity rises from 0 to 8 m/s in 4 s, stays at 8 m/s for 6 s, then falls to 0 in 2 s. Find total distance.
- Acceleration phase: ½ × 4 × 8 = 16 m (triangle).
- Cruise phase: 6 × 8 = 48 m (rectangle).
- Deceleration: ½ × 2 × 8 = 8 m (triangle).
- Total = 16 + 48 + 8 = 72 m.
Tangent for instantaneous values
- Instantaneous speed on a d-t graph = gradient of the tangent at that time.
- Instantaneous acceleration on a v-t graph = gradient of the tangent at that time.
To draw a tangent: align a ruler at the point so its slope matches the curve.
⚠Common mistakes
- Reading distance directly from a v-t graph — it's the area, not the height.
- Reading speed directly from a d-t graph — it's the gradient.
- Forgetting that horizontal lines on v-t mean constant velocity (not zero motion).
- Confusing positive and negative velocities (direction).
AI-generated · claude-opus-4-7 · v3-deep-physics