Trains, Boats & Streams

Scenario

Whether it's two Vande Bharat trains crossing each other or a Swiggy Instamart rider biking against heavy wind, relative speed dictates how fast they close the gap.

Mental model

Relative speed: Opposite directions ADD up, same directions SUBTRACT. Boats: Downstream ADDS, upstream SUBTRACTS.

For trains, always add their lengths as the total distance to cross. For boats, the water's speed physically pushes or pulls the boat.

Formula

Trains crossing each other: Relative Speed = Speed1 + Speed2 (opposite) or |Speed1 - Speed2| (same direction). Time = (L1 + L2) / Relative Speed. Train crossing a stationary object (pole, person): Time = L_train / Speed_train. Train crossing a platform/bridge: Time = (L_train + L_platform) / Speed_train. Boats downstream: Speed = Boat_speed + Stream_speed (B + W). Upstream: Speed = Boat_speed - Stream_speed (B - W). Boat speed in still water = (Downstream + Upstream) / 2. Stream speed = (Downstream - Upstream) / 2. Speed conversion: x m/s = x * 18/5 km/hr. x km/hr = x * 5/18 m/s.

For a train crossing a stationary pole, L2 = 0. For a stationary platform/bridge, the platform has no speed but has length. Add L_train + L_platform as the total distance. Always convert units consistently (m/s vs km/hr) before computing time.

Worked example

Problem: A train 240m long crosses a platform 160m long in 20 seconds. What is the speed of the train in km/hr?

1. Total distance = length of train + length of platform = 240 + 160 = 400m.
2. Time = 20 seconds. Speed = Distance / Time = 400 / 20 = 20 m/s.
3. Convert to km/hr: 20 x (18/5) = 20 x 3.6 = 72 km/hr.
4. Sanity-check: 72 km/hr = 72000m / 3600s = 20 m/s. Correct.

Answer: Speed = 72 km/hr.

Problem: A boat goes downstream at 15 km/hr and upstream at 9 km/hr. Find the speed of the boat in still water and the speed of the stream.

1. Boat speed in still water = (Downstream + Upstream) / 2 = (15 + 9) / 2 = 24 / 2 = 12 km/hr.
2. Stream speed = (Downstream - Upstream) / 2 = (15 - 9) / 2 = 6 / 2 = 3 km/hr.
3. Sanity-check: Downstream = 12 + 3 = 15. Upstream = 12 - 3 = 9. Correct.

Answer: Boat speed = 12 km/hr, Stream speed = 3 km/hr.

Shortcuts

• Finding Boat and Stream Speed: If downstream speed is D and upstream is U: Boat speed = (D + U)/2. Stream speed = (D - U)/2. (Downstream 10 km/hr, Upstream 6 km/hr. Boat = (10+6)/2 = 8 km/hr. Stream = (10-6)/2 = 2 km/hr.)

Common mistakes

• Subtracting Train Lengths: Thinking that because trains move in the same direction, you subtract their lengths. Always ADD the lengths - both trains must completely pass each other.
• Confusing Relative Speed Direction: Trains moving in opposite directions: ADD speeds. Trains moving same direction: SUBTRACT speeds (faster minus slower). Boats downstream: ADD stream speed. Boats upstream: SUBTRACT stream speed.

Glossary

relative

How fast something is moving compared to something else that is also moving.

downstream

Moving in the same direction that the water is flowing.

upstream

Moving against the flow of the water, making it harder and slower.

Recall questions

• Two trains of length 100m and 150m cross each other. What is the total distance covered?
• A boat goes 15 km/hr downstream and 5 km/hr upstream. What is the speed of the stream?
• If two trains move in opposite directions at 40 km/hr and 60 km/hr, what is their relative speed?

Questions & answers

A 150m long train crosses a pole in 15 seconds. What is the speed of the train in km/hr?

36 km/hr

Approach: Speed = 150m / 15s = 10 m/s. Convert: 10 x (18/5) = 36 km/hr.

A man can row 8 km/hr in still water. If the river runs at 2 km/hr and it takes him 4 hours to row to a place and back, how far is the place?

15 km

Approach: Downstream = 8+2=10 km/hr. Upstream = 8-2=6 km/hr. Let distance = D. D/10 + D/6 = 4. 3D/30 + 5D/30 = 4. 8D/30 = 4. D = 15 km.

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