GRADE 9 Mathematics MEASUREMENTS – TIME,DISTANCE AND SPEED Notes
Mathematics — Measurements
Subtopic: Time, Distance and Speed (Age group: 14 — Kenyan context)
1. Key concepts
- Distance (d) — how far an object travels. Common units: metres (m), kilometres (km).
- Time (t) — how long the travel takes. Common units: seconds (s), minutes (min), hours (h).
- Speed (v) — how fast an object moves. It is distance divided by time.
Formula (simple):
v = d ÷ t
Where v = speed, d = distance, t = time.
2. Units and conversions
- 1 km = 1000 m
- 1 h = 60 min = 3600 s
- Common speed units: km/h (kilometres per hour), m/s (metres per second)
- To convert km/h to m/s: multiply by 1000/3600 = 1/3.6. So, m/s = (km/h) ÷ 3.6
- To convert m/s to km/h: multiply by 3.6
Quick conversions (useful):
18 km/h = 5 m/s (since 18 ÷ 3.6 = 5). 72 km/h = 20 m/s.
3. Types of speed
- Uniform speed — speed is constant (straight line on a distance-time graph).
- Non-uniform speed — speed changes; can accelerate or slow down.
- Average speed — total distance ÷ total time (use when speed varies).
4. Worked examples (Kenyan context)
Example 1 — Car between towns
A car travels from Nakuru to Nairobi, a distance of 160 km, in 2 hours. What is its speed in km/h and m/s?
Speed = distance ÷ time = 160 km ÷ 2 h = 80 km/h.
Convert to m/s: 80 ÷ 3.6 ≈ 22.22 m/s.
Example 2 — Matatu trip with stops (average speed)
A matatu travels 30 km in 30 minutes, but there are stops and traffic. What is its average speed in km/h?
Convert time: 30 minutes = 0.5 h. Average speed = 30 km ÷ 0.5 h = 60 km/h.
Note: Actual instantaneous speed may be higher between stops, but the average is 60 km/h.
Example 3 — Walking and cycling (multi-segment)
Juma walks 2 km at 5 km/h and then cycles 6 km at 20 km/h. What is his average speed for the whole trip?
Time walking = distance ÷ speed = 2 ÷ 5 = 0.4 h. Time cycling = 6 ÷ 20 = 0.3 h. Total distance = 8 km. Total time = 0.4 + 0.3 = 0.7 h.
Average speed = total distance ÷ total time = 8 ÷ 0.7 ≈ 11.43 km/h.
5. Distance–Time Graph (visual)
On a distance–time graph:
- Horizontal axis (x-axis) = time
- Vertical axis (y-axis) = distance
- Slope of the line = speed. Steeper line = faster speed.
In the example graph, slope = 120 km ÷ 3 h = 40 km/h.
6. Worked conversion example
A bus moves at 54 km/h. Find its speed in m/s.
m/s = 54 ÷ 3.6 = 15 m/s.
7. Practice questions (with answers)
- Calculate the speed of a bicycle that covers 15 km in 1.5 hours. (Answer: 10 km/h)
- A motorcycle goes 72 km in 1.2 hours. Find its speed in km/h and m/s. (Answer: 60 km/h; 16.67 m/s)
- Bus A travels 200 km in 2.5 hours. Bus B travels the same distance in 3 hours. Which is faster and by how much? (Answer: Bus A; speeds 80 km/h and 66.67 km/h; difference ≈13.33 km/h)
- On a school trip, a van drives 90 km. It goes 40 km/h for the first 30 km and then 60 km/h for the remaining 60 km. What is the average speed for the whole trip? (Worked answer below.)
Answer to question 4 (click to expand)
Time for first 30 km = 30 ÷ 40 = 0.75 h. Time for next 60 km = 60 ÷ 60 = 1 h. Total distance = 90 km. Total time = 1.75 h. Average speed = 90 ÷ 1.75 ≈ 51.43 km/h.
8. Tips for exams and practical use
- Always check and convert units before using the formula v = d ÷ t.
- When using time in hours, convert minutes to hours (e.g., 30 min = 0.5 h).
- For multi-segment trips, find each time then add to get total time for average speed.
- Use distance–time graphs to compare speeds: steeper = faster.