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.

Worked Examples

**Example 1 (Basic):** A car travels 150 km in 3 hours. Find its speed. **Solution:** Speed = Distance / Time = 150 / 3 = 50 km/h.

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**Example 2 (Unit Conversion and Train):** A train 120 m long crosses a pole in 6 seconds. Find its speed in km/h. **Solution:** Distance = length of train = 120 m (a pole has negligible length). Time = 6 s. Speed in m/s = 120 / 6 = 20 m/s. Convert to km/h: 20 × (18/5) = 72 km/h.

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**Example 3 (Trains in Opposite Directions):** Two trains of lengths 100 m and 150 m travel towards each other at 54 km/h and 72 km/h. How long do they take to cross each other? **Solution:** Relative speed = 54 + 72 = 126 km/h (opposite directions). Convert to m/s: 126 × (5/18) = 35 m/s. Total distance to cover = 100 + 150 = 250 m. Time = 250 / 35 = 50/7 ≈ 7.14 seconds.

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**Example 4 (Boats and Streams):** A boat's speed in still water is 15 km/h. Stream speed is 3 km/h. Find downstream and upstream speeds, and time to travel 36 km downstream. **Solution:** Downstream speed = 15 + 3 = 18 km/h. Upstream speed = 15 − 3 = 12 km/h. Time downstream = 36 / 18 = 2 hours.

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**Example 5 (Average Speed):** A cyclist covers 60 km at 20 km/h and the next 60 km at 30 km/h. Find average speed. **Solution:** Time for first part = 60 / 20 = 3 hours. Time for second part = 60 / 30 = 2 hours. Total distance = 120 km; total time = 5 hours. Average speed = 120 / 5 = 24 km/h. (Note: (20 + 30)/2 = 25 would be **wrong**.)

Common Mistakes

1. **Averaging speeds arithmetically** — Students compute (S₁ + S₂)/2 instead of Total Distance / Total Time. Always use the definition of average speed. **Fix:** Calculate total distance and total time separately, then divide.

2. **Forgetting unit conversion** — Mixing km/h and m/s in the same calculation, or applying the wrong factor. **Fix:** Convert all speeds to the same unit before computing. Remember 1 km/h = 5/18 m/s, not 1/3.6.

3. **Adding train lengths incorrectly** — For a train crossing a platform, students use only train length or only platform length. **Fix:** Train + platform when crossing a platform; train length alone when crossing a pole; sum of both train lengths when two trains cross.

4. **Confusing upstream and downstream** — Assigning boat + stream as upstream speed. **Fix:** Downstream = boat speed + stream (moving with current). Upstream = boat speed − stream (against current).

5. **Relative speed sign error** — Subtracting speeds when objects move towards each other, or adding when they move in the same direction. **Fix:** Opposite directions → add speeds. Same direction → subtract speeds.

Quick Reference

• **D = S × T** — All TSD problems reduce to this. Isolate the unknown.
• **1 km/h = 5/18 m/s** — Memorize this conversion factor cold.
• **Opposite directions → add speeds; same direction → subtract speeds.**
• **Average speed = Total Distance ÷ Total Time** — never the arithmetic mean of two speeds unless distances are identical.
• **Trains crossing: pole → train length; platform → train + platform; another train → sum of both lengths.**
• **Boats: downstream = still + stream; upstream = still − stream; still = (down + up)/2; stream = (down − up)/2.**