Time-Speed-Distance
Time-speed-distance is one equation used four different ways: distance equals speed times time. Every variant — trains passing, boats against a current, two people meeting — is the same equation with a twist added to speed or distance.
Mental model
Everything is one triangle: Distance = Speed x Time. Convert units FIRST — kilometres per hour to metres per second is multiply by 5/18, and the reverse is multiply by 18/5 — because the classic trap is a perfectly correct method applied to mismatched units. Then remember the two relative-speed rules: same direction subtract the speeds, opposite direction add them. For trains, the distance covered is the train's own length plus the length of whatever it is crossing; a pole has no length, a platform does.
Associations
A short trigger on the left, everything it pulls with it on the right.
Same direction subtracts speed; opposite direction adds it.
Two objects moving the same way close the gap at the difference of their speeds (relative speed); moving toward each other, they close it at the sum. This single idea covers 'two trains', 'two people walking', and 'boat with or against current' questions.
A train passing a pole covers its own length; passing a platform covers its length plus the platform's.
The distance in a 'train problem' is almost always a length, not a route distance — get the distance right and the rest is the plain distance-speed-time equation.
Things to remember
The night-before list. Short enough to recall cold.
- 1Distance = Speed x Time; the same equation rearranges to Speed = Distance / Time and Time = Distance / Speed.
- 2To convert km/h to m/s, multiply by 5/18; to convert m/s to km/h, multiply by 18/5.
- 3A boat's downstream speed is (boat speed + current speed); its upstream speed is (boat speed - current speed).
- 4When two people start at the same time from opposite ends and move toward each other, they meet after time equal to the total distance divided by the sum of their speeds.
In detail
- Speed conversion: kilometres per hour to metres per second, multiply by 5/18; metres per second to kilometres per hour, multiply by 18/5.
- Relative speed: moving in the same direction, subtract the speeds; moving in opposite directions, add them.
- A train crossing a pole covers only its own length; crossing a platform or another train it covers the sum of both lengths.
- Boats: downstream speed = boat speed + stream speed; upstream speed = boat speed - stream speed. Hence boat speed = (downstream + upstream)/2 and stream speed = (downstream - upstream)/2.
- For equal distances at two different speeds, the average speed is the harmonic mean 2xy/(x+y), never the simple arithmetic average.
- At constant distance, speed and time are inversely proportional: if speed becomes 3/4, time becomes 4/3 of the original.
Where people slip
- When a train crosses a platform, the distance covered is the train's length PLUS the platform's length — using the train's length alone is the most common error.
Check yourself
Nothing here is scored — it will not change your mastery figures.
A train 200 m long crosses a platform 300 m long in 25 seconds. What is the speed of the train in m/s?
From the question bank
A car travels at a speed of 60 km/h for 3 hours. What distance does it cover?
From the question bank
A bus covers a distance of 150 km at a speed of 50 km/h. How much time does it take?
Books to read
- Quantitative Aptitude for Competitive Examinations — R S Aggarwal
- Fast Track Objective Arithmetic — Arihant
Official sources
Hand-checked official domains only — the notes above carry no links of their own.