The basic relation
Distance = Speed × Time
Speed = Distance ÷ Time and Time = Distance ÷ Speed. Keep the units consistent: km with hours, metres with seconds.
| Conversion | Multiply by | Example |
|---|---|---|
| km/h → m/s | 5/18 | 72 km/h = 20 m/s |
| m/s → km/h | 18/5 | 25 m/s = 90 km/h |
- At constant distance, speed and time are inversely proportional. Going at 3/4 of the usual speed takes 4/3 of the usual time; if that makes the journey 20 minutes late, the usual time is 60 minutes.
- Speeds in ratio a : b over the same distance give times in ratio b : a.
Average Speed
Average speed
Average speed = Total distance ÷ Total time
For two equal distances at speeds x and y: 2xy/(x + y). At 40 and 60 km/h the average is 4800/100 = 48 km/h, not 50. For equal times at two speeds, the average is simply (x + y)/2. Halts count in the total time.
Relative Speed
- Moving in the same direction: relative speed = difference of speeds. Opposite directions: relative speed = sum of speeds.
- Time to meet or catch up = gap ÷ relative speed. A train leaving at 6 am at 60 km/h is caught by one leaving at 8 am at 90 km/h after a 120 km gap closes at 30 km/h: 4 hours, at noon.
- Two bodies starting together towards each other meet after (distance ÷ sum of speeds). Anything shuttling between them for that time covers its own speed × that time.
Trains
| Train crosses | Distance covered |
|---|---|
| A pole, a signal post or a standing man | Its own length |
| A platform, bridge or tunnel | Own length + platform length |
| Another train | Sum of both lengths, at the relative speed |
| A man walking | Own length, at the relative speed |
Worked example
Trains of 200 m and 300 m approach each other at 60 and 90 km/h. Relative speed = 150 km/h = 125/3 m/s; distance = 500 m; time = 500 × 3/125 = 12 s.
If a train crosses a pole in t₁ seconds and a platform of length P in t₂ seconds, the extra time (t₂ − t₁) is spent covering P, so speed = P/(t₂ − t₁) and length = speed × t₁.
Boats and Streams
Downstream and upstream
D = b + s, U = b − s; b = (D + U)/2, s = (D − U)/2
b = speed of the boat in still water, s = speed of the stream. Upstream 4 km/h and downstream 6 km/h give a boat speed of 5 km/h and a stream of 1 km/h.
Circular Tracks and Races
- Two runners from the same point on a track of length L meet for the first time after L ÷ relative speed: the sum of speeds if they run in opposite directions, the difference if in the same direction.
- They are next together at the starting point after the LCM of their lap times.
- 'A beats B by x m in a race of d m' means when A finishes d, B has run d − x, so their speeds are in the ratio d : (d − x).
Common Mistakes
Mixing km/h with metres and seconds; averaging the two speeds for equal distances; forgetting to add the platform or the other train's length; and adding speeds when both move the same way.