Sketch, plot and interpret distance-time and speed-time graphs

1.2 Motion – Distance‑time & Speed‑time graphs

What is Distance and Speed?

Distance ($d$) is the total length travelled, measured in metres (m). Speed ($v$) is how fast you are moving, measured in metres per second (m s⁻¹). Analogy: Think of distance as the number of steps you take on a walk and speed as how quickly you take those steps. 🚶‍♂️

Distance‑time Graphs

A distance‑time graph plots distance (y‑axis) against time (x‑axis). Key features:

• Slope = speed: $v = \frac{Δd}{Δt}$
• Intercept = starting distance (usually 0)
• Shape tells you if speed is constant, increasing or decreasing.

Example 1: Constant Speed

A car travels at a constant 20 m s⁻¹ for 5 s. Distance after 5 s: $d = 20 \times 5 = 100$ m. The graph is a straight line with slope 20 m s⁻¹. 📈

Example 2: Accelerating Motion

A runner starts from rest and accelerates at 2 m s⁻² for 4 s. Distance: $d = \tfrac{1}{2} a t^2 = \tfrac{1}{2} \times 2 \times 4^2 = 16$ m. Graph is a curve (parabola) opening upwards. 🏃‍♂️

Speed‑time Graphs

A speed‑time graph plots speed (y‑axis) against time (x‑axis). Key features:

• Area under the curve = distance travelled.
• Slope = acceleration.
• Shape shows how speed changes.

Example 3: Constant Speed

Speed = 15 m s⁻¹ for 10 s. Distance = area = $15 \times 10 = 150$ m. Graph is a horizontal line. 📊

Example 4: Deceleration

A cyclist slows from 12 m s⁻¹ to 0 in 6 s. Area = $\tfrac{1}{2} \times 12 \times 6 = 36$ m. Graph is a straight line descending to zero. 🛑

Interpreting Graphs

• Reading slope gives instantaneous speed (distance‑time) or acceleration (speed‑time).
• Reading area under a speed‑time graph gives total distance.
• Flat sections mean constant speed.
• Curved sections mean changing speed.

Practice Questions

1. A train travels 30 m s⁻¹ for 2 min. Sketch the distance‑time graph and calculate the total distance.
2. During a 5 s interval, a car’s speed increases from 10 m s⁻¹ to 20 m s⁻¹. Draw the speed‑time graph and find the distance travelled.
3. Explain why the area under a speed‑time graph is equal to the distance travelled.

Examination Tips

• Always label axes with units.
• Check that the graph starts at the origin unless stated otherwise.
• When asked for distance, look for the area under a speed‑time graph.
• For speed from a distance‑time graph, calculate the slope between two points.
• Use clear, neat sketches; examiners appreciate legible graphs.