Trains, boats and streams for AFCAT: lengths, relative speed and upstream–downstream

Trains and boats are speed, time and distance with one extra rule each. For a train, the distance includes the train's own length. For a boat, the water adds to or takes away from the boat's speed. Get those two rules right and the rest is the ordinary formula, distance = speed × time.

These questions reward careful set-up more than clever arithmetic. Most wrong answers come from forgetting a length, using the wrong relative speed or leaving speed in km/h while lengths are in metres. This guide covers each case and the reason behind it.

Trains: what distance is covered?

A train has "crossed" something when its back end clears it. So the front of the train has to travel its own length plus the length of whatever it is crossing.

Lengths are nearly always in metres, so convert speeds to m/s by multiplying by 5/18. (54 km/h = 15 m/s, 72 km/h = 20 m/s, 90 km/h = 25 m/s.)

The meeting-time rule

Two trains start at the same time from P and Q towards each other. After meeting, they take t₁ and t₂ hours to reach Q and P respectively. Then

speed of first : speed of second = √t₂ : √t₁

Why: after meeting, each train covers the stretch the other covered before meeting. If they met after t hours, S₁ × t₁ = S₂ × t and S₂ × t₂ = S₁ × t. Dividing one by the other gives S₁²/S₂² = t₂/t₁.

So if the trains take 9 hours and 16 hours after meeting, their speeds are in the ratio √16 : √9 = 4 : 3.

Boats and streams

A boat moving with the current gets a push; against it, a drag.

• Downstream speed = boat speed + stream speed.
• Upstream speed = boat speed − stream speed.

Add and subtract these two lines and you get the two formulas that solve most questions:

• Boat speed in still water = (downstream + upstream) ÷ 2.
• Stream speed = (downstream − upstream) ÷ 2.

For a round trip, add the two times separately. A boat at 8 km/h in still water, on a stream of 2 km/h, takes 15 ÷ 10 = 1.5 hours to go 15 km downstream and 15 ÷ 6 = 2.5 hours to come back: 4 hours in all. Its average speed is 30 ÷ 4 = 7.5 km/h, less than 8, because it spends longer going slowly.

Worked examples

Example 1 (a pole). A 150 m train at 54 km/h passes a pole. How long does it take?

• 54 km/h = 15 m/s.
• 150 ÷ 15 = 10 seconds.

Example 2 (pole and platform). A train passes a pole in 12 seconds and a 240 m platform in 24 seconds. Find its length and speed.

• The extra 12 seconds are spent covering the platform's 240 m, so speed = 240 ÷ 12 = 20 m/s = 72 km/h.
• Length = 20 × 12 = 240 m.

Example 3 (two trains, opposite directions). Trains 120 m and 180 m long run in opposite directions at 50 km/h and 40 km/h. How long do they take to cross each other?

• Relative speed = 90 km/h = 25 m/s.
• Distance = 120 + 180 = 300 m.
• Time = 300 ÷ 25 = 12 seconds.

Example 4 (a moving person). A 110 m train at 60 km/h passes a person walking at 6 km/h in the opposite direction. How long does it take?

• Relative speed = 66 km/h = 66 × 5/18 = 55/3 m/s.
• Time = 110 ÷ (55/3) = 6 seconds.

Example 5 (boat and stream). A boat goes 24 km downstream in 2 hours and returns upstream in 3 hours. Find the speeds of the boat and the stream.

• Downstream = 12 km/h, upstream = 8 km/h.
• Boat = (12 + 8) ÷ 2 = 10 km/h; stream = (12 − 8) ÷ 2 = 2 km/h.

Example 6 (ratio from times). A boat takes twice as long to go upstream as to cover the same distance downstream. Find the ratio of boat speed to stream speed.

• Same distance, so downstream speed is twice the upstream speed: b + s = 2(b − s).
• b = 3s, so the ratio is 3 : 1.

Common mistakes

• Leaving speed in km/h while lengths are in metres. Always convert to m/s.
• Forgetting the platform in "crosses a platform" questions, or adding it for a signal post.
• Using the sum of speeds for trains going the same way. Same direction subtracts.
• Mixing boat and stream speeds. The still-water speed is the average of downstream and upstream, not either one.

Practice set

1. A 240 m train at 72 km/h passes a pole. How long does it take?
2. A boat's speed in still water is 15 km/h and the stream flows at 3 km/h. How long does it take to go 36 km downstream?
3. Trains 100 m and 150 m long run in the same direction at 60 km/h and 42 km/h. How long does the faster take to pass the slower?
4. A 300 m train crosses a 200 m bridge in 25 seconds. Find its speed in km/h.
5. A boat goes 30 km upstream in 5 hours and 44 km downstream in 4 hours. Find the speeds of the boat and the stream.
6. Two trains, each 120 m long, pass a pole in 10 seconds and 15 seconds. How long will they take to cross each other in opposite directions?
7. A rower's speed in still water is 5 km/h and the stream flows at 1 km/h. A trip to a point and back takes 1 hour. How far is the point?
8. A 125 m train passes a person running at 5 km/h in the same direction in 10 seconds. Find the train's speed.

Answers:

1. 12 seconds. 72 km/h = 20 m/s, and 240 ÷ 20.
2. 2 hours. Downstream 18 km/h, and 36 ÷ 18.
3. 50 seconds. Relative 18 km/h = 5 m/s. Distance 250 m, and 250 ÷ 5.
4. 72 km/h. 500 m in 25 s = 20 m/s.
5. Boat 8.5 km/h, stream 2.5 km/h. Upstream 6, downstream 11. (11 + 6) ÷ 2 and (11 − 6) ÷ 2.
6. 12 seconds. Speeds 12 m/s and 8 m/s, relative 20 m/s. Distance 240 m.
7. 2.4 km. d/6 + d/4 = 1, so 5d/12 = 1 and d = 12/5.
8. 50 km/h. Relative speed = 125 ÷ 10 = 12.5 m/s = 45 km/h, and 45 + 5 = 50.

What to do next

• Copy the train table into your formula notebook and fill it from memory once a day for a week.
• Practise ten train questions and ten boat questions, writing "length" and "speed" before each one.
• Revise speed, time and distance for average and relative speed, then move on to mixtures and alligation.