The Bus and the Overtaking Train
A bus leaves Sylhet at 7:00 a.m. and travels toward Dhaka, 240 km away, at a steady 40 km/h. A train on a parallel track leaves Sylhet at 8:00 a.m. and travels the same route at 60 km/h. The train must pass through a 300 m long tunnel and also overtake a 200 m long goods train moving in the same direction at 30 km/h. The passenger train itself is 150 m long. The bus makes a 20-minute breakfast stop at 9:00 a.m. A passenger on the bus claims the train will catch the bus before the bus reaches Dhaka. The bus driver disagrees.
1. Determine whether the train catches the bus before Dhaka, accounting for the bus's 20-minute stop.
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By 8:00 a.m. the bus has travelled 1 hour at 40 km/h = 40 km. Remaining gap = 40 km.
Relative speed of train vs bus = 60 − 40 = 20 km/h. Without any stop, catch-up time = 40/20 = 2 hours, i.e. at 10:00 a.m.
But the bus stops 20 minutes (1/3 hour) at 9:00 a.m. During that stop the train closes an extra 60 × 1/3 = 20 km relative to the bus's moving position.
Simpler check: at 9:00 a.m., bus has travelled 40 + 40 = 80 km (position 80 km). Train left at 8:00, so by 9:00 it has gone 60 km. Gap at 9:00 = 80 − 60 = 20 km. Bus then stops 20 min; train covers 60 × 1/3 = 20 km, exactly closing the gap. So the train catches the bus at 9:20 a.m., at position 80 km.
Dhaka is at 240 km, so the catch happens well before Dhaka. The passenger is right; the driver is wrong.
2. Find the time the passenger train takes to completely pass through the 300 m tunnel.
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To completely pass through a tunnel, the train must travel its own length plus the tunnel length.
Total distance = 300 m + 150 m = 450 m.
Train speed = 60 km/h = 60 × (1000/3600) m/s = 60000/3600 = 50/3 m/s ≈ 16.67 m/s.
Time = distance/speed = 450 ÷ (50/3) = 450 × 3/50 = 27 seconds.
So the train takes 27 seconds to fully clear the tunnel.
3. Find the time the passenger train takes to completely overtake the 200 m goods train moving at 30 km/h in the same direction.
Read the worked answer
For same-direction overtaking, use relative speed and total length.
Relative speed = 60 − 30 = 30 km/h = 30 × (5/18) m/s = 25/3 m/s ≈ 8.33 m/s.
Total length to cover = 150 m + 200 m = 350 m.
Time = 350 ÷ (25/3) = 350 × 3/25 = 1050/25 = 42 seconds.
So the passenger train needs 42 seconds to completely overtake the goods train. (Note: had they moved in opposite directions, relative speed would be 90 km/h and time would be 14 seconds - direction matters.)