Time, Speed and Distance — Study Notes

Overview

Time, Speed and Distance (TSD) forms the backbone of motion-based word problems in competitive mathematics. In SOF IMO, this topic tests your ability to translate real-world scenarios—trains crossing platforms, boats rowing upstream, cyclists meeting on roads—into clean mathematical equations. Mastery here requires fluency with the fundamental relationship **Distance = Speed × Time** and its rearrangements, plus pattern recognition across three classic sub-types: relative motion (trains), upstream/downstream (boats and streams), and average speed calculations.

Expect 2–4 direct questions in the Everyday Mathematics section, often worded as multi-step problems involving unit conversions (km/h to m/s), opposite-direction motion, or time-gap scenarios. Strong performance demands quick mental calculation, clear formula recall, and careful reading—students frequently lose marks by mixing up "towards" versus "away" or forgetting to add train lengths. The Achievers Section may combine TSD with ratio, percentage, or data interpretation, so practice linking this topic to others.

Key Concepts

• **Core Formula**: Distance (D) = Speed (S) × Time (T). Any one can be found if the other two are known. Speed = D/T, Time = D/S.
• **Unit Conversion**: Always match units. 1 km/h = 5/18 m/s; to convert km/h to m/s, multiply by 5/18. To convert m/s to km/h, multiply by 18/5.
• **Relative Speed (Same Direction)**: When two objects move in the same direction, their relative speed is the difference: S₁ − S₂.
• **Relative Speed (Opposite Direction)**: When two objects move towards each other, their relative speed is the sum: S₁ + S₂.
• **Average Speed**: Average speed ≠ arithmetic mean of speeds. Use **Average Speed = Total Distance / Total Time**. Never average two speeds directly unless distances are equal.
• **Trains Crossing**: When a train crosses a platform, pole, or another train, add relevant lengths to the distance. Crossing a pole uses train length; crossing a platform uses train length + platform length; crossing another train uses sum of both train lengths.
• **Boats and Streams**: Downstream speed = (speed in still water) + (stream speed). Upstream speed = (speed in still water) − (stream speed). Speed in still water = (downstream + upstream)/2. Stream speed = (downstream − upstream)/2.
• **Meeting and Crossing**: If two objects start simultaneously from two points and move towards each other, time to meet = (distance between them) / (sum of speeds).

Formulas / Key Facts

1. **D = S × T** — Core relation. Speed in consistent units (km/h or m/s). 2. **1 km/h = 5/18 m/s** — Multiply km/h by 5/18 to get m/s; multiply m/s by 18/5 to get km/h. 3. **Relative speed (opposite)** = S₁ + S₂ — Use when objects approach each other. 4. **Relative speed (same direction)** = |S₁ − S₂| — Use when one overtakes the other. 5. **Average Speed** = Total Distance / Total Time — Not (S₁ + S₂)/2 unless both legs cover equal distance. 6. **Train crossing a pole/signal** — Distance = Length of train. 7. **Train crossing a platform/bridge** — Distance = Length of train + Length of platform. 8. **Two trains crossing each other** — Distance = Sum of lengths of both trains. 9. **Downstream speed (D)** = B + R, **Upstream speed (U)** = B − R, where B = boat speed in still water, R = stream speed. 10. **Speed in still water** = (D + U)/2; **Stream speed** = (D − U)/2.