Speed, Distance and Time
1. Basic Relationships
The three quantities are related by: Speed = Distance / Time
From this we derive:
- Distance = Speed × Time
- Time = Distance / Speed
Units
| Quantity | Common Units |
|---|---|
| Speed | m/s, km/h, cm/s |
| Distance | m, km, cm |
| Time | seconds (s), hours (h), minutes (min) |
Key Conversion
km/h to m/s: Multiply by 5/18. m/s to km/h: Multiply by 18/5.
Example: 72 km/h = 72 × 5/18 = 20 m/s. Example: 15 m/s = 15 × 18/5 = 54 km/h.
2. Average Speed
Average Speed = Total Distance Travelled / Total Time Taken
Important
Average speed is NOT simply the arithmetic mean of individual speeds (unless time is equal).
Worked Example (ICSE 2024, 3 marks)
'A car travels from A to B at 40 km/h and returns from B to A at 60 km/h. Find the average speed.'
Solution: Let distance from A to B = d km. Time A→B = d/40 h. Time B→A = d/60 h. Total distance = 2d km. Total time = d/40 + d/60 = (3d + 2d)/120 = 5d/120 = d/24 h. Average speed = 2d / (d/24) = 2d × 24/d = 48 km/h.
3. Speed When Distance is Equal
For two equal distances travelled at speeds u and v: Average speed = 2uv/(u + v)
This is the formula used in the example above.
4. Word Problems
Problem Type 1: Finding Speed
'A train travels 240 km in 4 hours. Find its speed in km/h and m/s.' Speed = 240/4 = 60 km/h. In m/s: 60 × 5/18 = 300/18 = 50/3 m/s = 16.67 m/s.
Problem Type 2: Finding Distance
'A cyclist travels at 12 km/h for 2 hours 30 minutes. Find the distance.' Time = 2.5 hours. Distance = 12 × 2.5 = 30 km.
Problem Type 3: Finding Time
'A bus travels at 54 km/h. How long will it take to cover 270 km?' Time = 270/54 = 5 hours.
Problem Type 4: Relative Speed (ICSE Focus)
'Two trains of length 100 m and 120 m run on parallel tracks at 72 km/h and 54 km/h. Find time to cross each other when moving in opposite directions.'
Solution: Relative speed = 72 + 54 = 126 km/h = 126 × 5/18 = 35 m/s. Total distance = 100 + 120 = 220 m. Time = 220/35 = 44/7 = 6.29 seconds.
5. Speed-Time Graphs
A DISTANCE-TIME graph shows how distance changes with time.
- Straight line upward → UNIFORM speed (constant).
- Horizontal line → STATIONARY (no movement).
- Curved line → NON-UNIFORM speed.
Finding Speed from Distance-Time Graph
Speed = SLOPE of the line = (Change in distance) / (Change in time).
6. Common Applications
Trains Crossing Objects
- Crossing a pole/light: distance = length of train.
- Crossing a platform: distance = length of train + length of platform.
- Crossing a bridge: distance = length of train + length of bridge.
Boats and Streams (Basic)
- Downstream speed = Speed in still water + Speed of stream.
- Upstream speed = Speed in still water - Speed of stream.
7. ICSE Exam Focus
| Topic | Marks | Frequency |
|---|---|---|
| Basic speed-distance-time problems | 2-3 marks | Very High |
| Unit conversions (km/h ↔ m/s) | 1-2 marks | Very High |
| Average speed | 3 marks | High |
| Train crossing problems | 3-4 marks | Medium |
| Distance-time graph | 2 marks | Low |
Common Mistakes
- Using mixed units (e.g., km with minutes) without converting.
- Average speed = average of speeds (WRONG — use total distance/total time).
- Forgetting to convert km/h to m/s in train problems.
- Time must be in hours when speed is in km/h (or convert consistently).
Unit Conversion Reference
| Speed (km/h) | Speed (m/s) |
|---|---|
| 18 | 5 |
| 36 | 10 |
| 54 | 15 |
| 72 | 20 |
| 90 | 25 |
| 108 | 30 |
Self-Test (5 Questions)
Q1. Convert 90 km/h to m/s. (1 mark)
- A) 20 m/s
- B) 25 m/s
- C) 30 m/s
- D) 15 m/s
Q2. 'A train 150 m long running at 54 km/h. How long will it take to cross a pole?' (2 marks)
Q3. 'A car travels 180 km in 3 hours. Find speed in m/s.' (2 marks)
Q4. 'A man walks at 5 km/h for 2 hours and then at 4 km/h for 3 hours. Find average speed.' (3 marks)
Q5. 'Two trains 200 m and 180 m long run at 54 km/h and 36 km/h on parallel tracks. Find time to cross when moving in the SAME direction.' (3 marks)
Answers
A1. B) 25 m/s. (90 × 5/18 = 25 m/s.) A2. 10 seconds. (54 km/h = 15 m/s. Time = 150/15 = 10 s.) A3. 50/3 = 16.67 m/s. (180 km / 3 h = 60 km/h. 60 × 5/18 = 50/3 m/s.) A4. 4.4 km/h. (D₁ = 10 km, D₂ = 12 km. Total = 22 km in 5 h. Avg = 22/5 = 4.4 km/h.) A5. 76 seconds. (Relative speed = 54 - 36 = 18 km/h = 5 m/s. Distance = 380 m. Time = 380/5 = 76 s.)
