Time, speed and distance
Distance, speed and time are linked by one relation, and nearly every problem is that relation applied with the right distance and the right speed. The usual work is choosing the speed that applies (relative speed, speed with or against a stream) and keeping the units consistent.
Formulas and when they apply
- Speed = Distance ÷ Time, so Distance = Speed × Time and Time = Distance ÷ SpeedApplies when: constant speed
- km/h to m/s: multiply by 5/18. m/s to km/h: multiply by 18/5
- Relative speed = sum of speeds if moving towards or away from each other, difference if moving in the same direction
- A train crossing a pole or a person covers its own length. Crossing a platform or another train, it covers the sum of the lengths
- Average speed = Total distance ÷ Total time. For equal distances at speeds a and b it is 2ab/(a + b)Applies when: equal distances only for the formula
- Downstream speed = u + v. Upstream speed = u − v. So u = (D + U)/2 and v = (D − U)/2Applies when: u is the boat's speed in still water, v the stream's speed
- On a circular track of length L, two runners starting together first meet after L ÷ relative speedApplies when: relative speed uses the sum for opposite directions and the difference for the same direction
Worked examples
Example 1. A train 240 m long crosses a pole in 12 seconds. How long does it take to cross a tunnel 360 m long at the same speed?
Solution. Speed = 240 ÷ 12 = 20 m/s. To cross the tunnel it must cover 240 + 360 = 600 m. Time = 600 ÷ 20 = 30 seconds.
Example 2. A boat covers 36 km downstream in 3 hours and returns the same 36 km upstream in 4 hours. Find the speed of the boat in still water.
Solution. Downstream speed = 36 ÷ 3 = 12 km/h. Upstream speed = 36 ÷ 4 = 9 km/h. Speed in still water = (12 + 9) ÷ 2 = 10.5 km/h. The stream speed is (12 − 9) ÷ 2 = 1.5 km/h.
Example 3. Two cyclists start together from the same point of a circular track 600 m long and ride in the same direction at 4 m/s and 6 m/s. After how many seconds do they first meet again?
Solution. Same direction, so the relative speed is 6 − 4 = 2 m/s. The faster cyclist must gain one full lap of 600 m. Time = 600 ÷ 2 = 300 seconds.
Shortcuts and their limits
- To go from km/h to m/s, multiply by 5/18. Memorise 36 km/h = 10 m/s and 72 km/h = 20 m/s as anchors.Does not apply when: Only for converting speeds. Convert before you combine speeds with lengths in metres.
- If the distance is fixed, time is inversely proportional to speed: speeds in the ratio 3 : 4 take times in the ratio 4 : 3.Does not apply when: Works only when the distance is the same in both cases.
- Average speed for equal distances is 2ab/(a + b). For 30 km/h and 60 km/h this is 2 × 30 × 60/90 = 40 km/h.Does not apply when: Not valid when the times are equal (then the average is the simple mean) or when the distances differ.
Common mistakes
- Averaging two speeds with a simple mean when the distances, not the times, are equal.
- Forgetting to convert km/h into m/s (or metres into kilometres) before dividing.
- Adding the speeds for a same-direction chase, or subtracting them for a head-on crossing.
- Using only the train's length when it crosses a platform, or adding a platform length when it crosses a pole.
- Using the still-water speed for both legs of a boat trip, forgetting the stream helps one way and hinders the other.
Independent practice made by EduVistaHQ with original questions. It is not an official examination paper, it is not affiliated with or endorsed by any examination body or company, and a practice score does not predict a real result.