Trains • Topic 3 of 3

Two Trains Crossing

When two trains pass each other, the total distance covered is the SUM of their lengths (each must clear the other from nose to tail), and the speed that governs the crossing is the relative speed. If they move in OPPOSITE directions, add the speeds — they approach each other faster, so the crossing is quick. If they move in the SAME direction, the faster overtakes the slower at the difference of speeds, so it takes longer. So time = (L₁ + L₂) ÷ relative speed. The same logic covers a train crossing a moving man: distance = the train’s length, speed = train ± man depending on direction. CAT’s favourite twist gives you BOTH a same-direction and an opposite-direction crossing time for the same two trains; set up two equations in the two speeds (sum and difference) and solve. Convert everything to m/s with 5/18 before adding.

✅ Solved examples

1. Two trains 150 m and 200 m long run towards each other at 54 km/h and 36 km/h. How long to cross?
Opposite ⇒ add: 54 + 36 = 90 km/h = 25 m/s. Distance = 150 + 200 = 350 m. Time = 350 ÷ 25 = 14 s.
2. A 120 m train at 90 km/h overtakes a 180 m train at 54 km/h going the same way. Time to fully overtake?
Same direction ⇒ subtract: 90 − 54 = 36 km/h = 10 m/s. Distance = 120 + 180 = 300 m. Time = 300 ÷ 10 = 30 s.
3. A 200 m train at 63 km/h crosses a man walking at 9 km/h in the opposite direction. Find the time.
Opposite ⇒ 63 + 9 = 72 km/h = 20 m/s. Distance = train length only = 200 m. Time = 200 ÷ 20 = 10 s.
4. Two trains of equal length cross each other in 10 s when moving towards each other and in 50 s when moving the same way. If each is 250 m long, find each speed (km/h).
Combined length = 500 m. Opposite: relative speed = 500/10 = 50 m/s = sum. Same: 500/50 = 10 m/s = difference. So speeds are (50+10)/2 = 30 and (50−10)/2 = 20 m/s ⇒ 108 km/h and 72 km/h.

✏️ Practice — try these, take hints as needed

1. Two trains 100 m and 150 m long move towards each other at 40 km/h and 50 km/h. Time to cross?
Opposite ⇒ add speeds.
90 km/h = 25 m/s.
Distance 250 ÷ 25.
10 s
2. A 160 m train at 80 km/h overtakes a 140 m train at 44 km/h (same direction). Overtaking time?
Same direction ⇒ subtract.
36 km/h = 10 m/s.
Distance 300 ÷ 10.
30 s
3. A 180 m train at 58 km/h passes a man running at 2 km/h in the opposite direction. Time?
Opposite ⇒ 58 + 2 = 60 km/h.
= 50/3 m/s.
Distance = 180 m only.
10.8 s
4. A 250 m train at 50 km/h overtakes a man walking at 5 km/h the same way. Time to pass him?
Same direction ⇒ 50 − 5 = 45 km/h.
= 12.5 m/s.
Distance = 250 m.
20 s
5. Two equal trains cross in 12 s (opposite) and 60 s (same direction). Combined length 600 m. Find the faster speed in km/h.
Sum = 600/12 = 50 m/s; diff = 600/60 = 10 m/s.
Faster = (50 + 10)/2 = 30 m/s.
30 × 18/5.
108 km/h

📝 Topic test — 8 questions

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