The core relation
Distance = speed × time. Keep units consistent.
To convert km/h to m/s, multiply by 5/18. To convert m/s to km/h, multiply by 18/5.
72 km/h × 5/18 = 20 m/s. 54 km/h = 15 m/s. 90 km/h = 25 m/s.
Trains
A train crossing something must cover its own length plus the length of what it crosses.
Crossing | Distance covered |
|---|---|
a pole, a person or a signal | length of the train |
a platform, bridge or tunnel | train + platform |
another train | sum of both lengths |
Example. A 200 m train at 72 km/h (20 m/s) crosses a 300 m platform in 500 ÷ 20 = 25 seconds.
Relative speed
Moving in opposite directions: add the speeds.
Moving in the same direction: subtract them.
Example. Trains of 120 m and 180 m run towards each other at 50 and 40 km/h. Relative speed = 90 km/h = 25 m/s, so they pass in 300 ÷ 25 = 12 s.
Chases. A thief with a 200 m head start runs at 10 km/h, chased at 12 km/h. The gap closes at 2 km/h = 2000 m per hour, so it takes 0.1 h = 6 minutes.
Meeting. Two people 90 km apart walking towards each other at 20 and 25 km/h meet after 90 ÷ 45 = 2 hours.
Boats and streams
With boat speed b in still water and stream speed s:
downstream speed = b + s
upstream speed = b − s
b = (downstream + upstream) / 2 and s = (downstream − upstream) / 2
Example. Downstream 20 km/h and upstream 12 km/h give b = 16 and s = 4 km/h.
Late and early
At a fraction k of the usual speed, the time becomes 1/k of the usual time.
Example. Walking at 4/5 of the usual speed makes someone 10 minutes late. The time is 5/4 of usual, so a quarter of the usual time is 10 minutes and the usual time is 40 minutes.
Average speed
For equal distances at a and b km/h: 2ab/(a + b). Otherwise: total distance ÷ total time.
Shortcuts and tricks
The standard way is distance = speed × time with units converted first. These shortcuts handle the common patterns directly.
km/h to m/s
Example: 90 km/h in m/s.
Standard way: 90 × 1000 ÷ 3600 = 25 m/s.
Shortcut: multiply by 5/18, or divide by 3.6: 90 ÷ 3.6 = 25 m/s.
Speed and time move in opposite ratios
Example: at 3/4 of the usual speed, a person is 20 minutes late. Find the usual time.
Standard way: let the usual time be t. The new time is 4t/3, so 4t/3 − t = 20 and t = 60 minutes.
Shortcut: speed × 3/4 means time × 4/3. The extra 1/3 of the usual time is 20 minutes, so the usual time is 60 minutes.
Boats: half the sum and half the difference
Example: downstream 14 km/h, upstream 8 km/h.
Standard way: b + s = 14 and b − s = 8, so b = 11 and s = 3.
Shortcut: boat = (14 + 8)/2 = 11 km/h, stream = (14 − 8)/2 = 3 km/h.
Two speeds over the same distance
Example: at 40 km/h a trip takes 30 minutes longer than at 50 km/h. Find the distance.
Standard way: d/40 − d/50 = 1/2, so d/200 = 1/2 and d = 100 km.
Shortcut: d = (product of speeds × time difference) ÷ speed difference = 40 × 50 × ½ ÷ 10 = 100 km.
Exam tips
Convert units before anything else.
Draw a quick line diagram for meeting and overtaking problems.
Time and speed are inversely proportional for a fixed distance.