Plotting and interpreting real-context graphs (kinematics and beyond)
These graphs model real situations: distance-time, speed-time, water filling a container, mobile phone tariffs, conversion graphs. The skill is reading the story of the graph: what's happening at each segment.
Distance-time graphs
x-axis: time. y-axis: distance from a starting point.
- Gradient = speed.
- Horizontal segment: stationary (speed 0).
- Steeper line = faster.
- Negative gradient = returning home (distance from start decreasing).
- Curve = changing speed (acceleration); gradient at a point = instantaneous speed.
Speed-time (velocity-time) graphs
x-axis: time. y-axis: speed.
- Gradient = acceleration.
- Horizontal segment: constant speed.
- Negative gradient: decelerating.
- Area under the graph = distance travelled.
✦Worked example— Worked example — distance-time
A walker leaves home and walks 4 km in 1 hour, rests for 30 min, then walks home in 1 hour.
- Segment 1: gradient = 4/1 = 4 → speed 4 km/h, going outward.
- Segment 2: horizontal at 4 km → resting.
- Segment 3: gradient = -4/1 = -4 → returning at 4 km/h.
Total time on the graph = 2.5 hours.
✦Worked example— Worked example — speed-time
A car accelerates from rest to 30 m/s in 10 s, holds 30 m/s for 20 s, then decelerates uniformly to rest in 5 s.
- Triangle on left, area = ½ × 10 × 30 = 150 m.
- Rectangle in middle, area = 20 × 30 = 600 m.
- Triangle on right, area = ½ × 5 × 30 = 75 m.
- Total distance = 150 + 600 + 75 = 825 m.
Container filling problems
A bottle has a narrow neck and wider body. As you pour water at a constant rate, height of water rises fast in the narrow neck (small cross-section), slow in the wide body. The time-vs-height graph is steeper where the container is narrower.
Conversion graphs
Plot quantity A on one axis, B on the other. To convert, find the value on one axis, draw a line up/across to the graph, then read off the other axis.
⚠Common mistakes— Common mistakes (examiner traps)
- Confusing distance-time gradient with speed-time gradient. Distance-time gives speed; speed-time gives acceleration.
- Forgetting the area under speed-time = distance. Common, costly slip.
- Reading negative-gradient segments of a distance-time graph as "going backwards in time" — they mean returning towards the starting point.
- Plotting straight lines through curved scenarios. A car accelerating uniformly gives a curved distance-time graph (parabola), but a straight speed-time graph.
- Forgetting units when stating gradients.
5 km/hnot5.
➜Try this— Quick check
A speed-time graph shows a straight line from (0, 0) to (8, 24) m/s. Find acceleration and distance.
- Acceleration = gradient = 24/8 = 3 m/s².
- Distance = ½ × 8 × 24 = 96 m.
AI-generated · claude-opus-4-7 · v3-deep-algebra