Quick revision notes · Mathematics 4024 · Chapter 11 of 35
Kinematics
Average speed, units, distance–time and speed–time graphs, acceleration, area as distance, and conversion graphs.
4 sectionsWorked examples
Common mistakes
At a glance
Average speed, units, distance–time and speed–time graphs, acceleration, area as distance, and conversion graphs.
1Speed, distance and time
- speed = distance ÷ time; distance = speed × time; time = distance ÷ speed.
- Average speed = total distance ÷ total time, including stops. It is not the average of the speeds.
- Convert units first: 45 minutes = 0.75 h; km/h ÷ 3.6 = m/s.
Formulas
speed = distance ÷ time
km/h → m/s÷ 3.6
m/s → km/h× 3.6
Worked example
A cyclist rides 30 km in 1 h, rests for 30 min, then rides 20 km in 1 h. Find the average speed.
- Total distance 50 km.
- Total time 2.5 h (the rest counts).
- 50 ÷ 2.5.
20 km/h
2Distance–time graphs
- Gradient = speed. Steeper = faster.
- Horizontal line = stationary. Line sloping down = returning towards the start.
- A curve means the speed is changing; use a tangent for the speed at an instant.
Worked example
A straight line on a distance–time graph goes from (0 s, 0 m) to (40 s, 200 m). Find the speed.
- Gradient = 200 ÷ 40.
5 m/s
3Speed–time graphs
- Gradient = acceleration (m/s²). Negative gradient = deceleration.
- Horizontal line = constant speed (not stopped!).
- Area under the graph = distance travelled. Split into triangles, rectangles and trapezia.
Formulas
acceleration = change in speed ÷ timegradient
distance = area under the graph
trapezium area = ½(a + b)h
Worked example
A bus accelerates from 0 to 12 m/s in 8 s, keeps 12 m/s for 20 s, then stops in 6 s. How far does it go?
- The shape is a trapezium with parallel sides 34 s and 20 s, height 12.
- Area = ½ × (34 + 20) × 12.
324 m
4Conversion graphs and rates
- A straight-line conversion graph through the origin means direct proportion: find the rate from one point.
- Use it to convert in either direction; for values off the graph, scale up.
- The gradient of any real-life graph is a rate (e.g. litres per minute, cost per kg).
Worked example
10 miles = 16 km. Convert 25 miles to km.
- 1 mile = 1.6 km.
- 25 × 1.6.
40 km
✗Common mistakes
| Don’t write | Write instead |
|---|---|
| 2 h 30 min = 2.3 h | 30 min = 0.5 h, so 2.5 h |
| Averaging the speeds of two parts | Average speed = total distance ÷ total time |
| Reading distance from a speed–time graph's axis | Distance is the AREA under it |
| A flat line on a speed–time graph means stopped | It means constant speed |
★In the exam
- Write the units: m/s, m/s², km/h.
- Split areas carefully and show each part.
- If the graph has different time units (minutes and hours), convert before calculating.