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Consider the following for the next three (03) items that follow:
Two trains A and B started from stations P and Q respectively towards each other. Train A started at 7 p.m. at a speed of 60 km/hr and train B started at 4 a.m. (next day) at a speed of 90 km/hr. The distance between the two stations P and Q is 800 km.

If the lengths of the two trains A and B are 400 m and 500 m respectively, then what is the time taken by them to cross each other?
 

This question was previously asked in
CDS 2 2024 Maths Question Paper (01-Sep-2024)
The correct answer is
21.6 seconds

Problem Analysis: Train Crossing Calculation

This problem involves calculating the time it takes for two trains, A and B, moving towards each other, to cross completely. We are given their starting times, speeds, the distance between their starting stations, and their lengths.

Given Information

  • Train A Speed (\(V_A\)): 60 km/hr
  • Train B Speed (\(V_B\)): 90 km/hr
  • Train A Start Time: 7 p.m.
  • Train B Start Time: 4 a.m. (the next day)
  • Distance between stations P and Q: 800 km
  • Length of Train A (\(L_A\)): 400 m
  • Length of Train B (\(L_B\)): 500 m

Step 1: Calculate Time Difference and Distance Covered by Train A

Train A starts earlier than Train B. We need to find out how far Train A travels before Train B begins its journey.

  • The time difference between Train A starting (7 p.m.) and Train B starting (4 a.m. next day) is 9 hours.
  • Distance covered by Train A during this time = Speed of A × Time difference
  • Let \(D_A\) be the distance covered by Train A.
  • \( D_A = V_A \times \text{time} = 60 \text{ km/hr} \times 9 \text{ hr} = 540 \text{ km} \)

Step 2: Determine Remaining Distance Between Trains

At 4 a.m., when Train B starts, Train A has already covered 540 km from station P towards station Q.

  • The initial distance between the trains when Train B starts is the total distance minus the distance covered by Train A.
  • Remaining Distance = Total Distance PQ - \(D_A\)
  • Remaining Distance = \(800 \text{ km} - 540 \text{ km} = 260 \text{ km}\)

Step 3: Calculate the Relative Speed of the Trains

Since the trains are moving towards each other, their speeds add up to give the relative speed at which they approach and cross each other.

  • Relative Speed (\(V_{rel}\)) = Speed of A + Speed of B
  • \( V_{rel} = V_A + V_B = 60 \text{ km/hr} + 90 \text{ km/hr} = 150 \text{ km/hr} \)

To calculate the crossing time accurately, especially since the lengths are given in meters, it's best to convert the relative speed to meters per second (m/s).

  • Conversion factor: 1 km/hr = \( \frac{1000}{3600} \) m/s = \( \frac{5}{18} \) m/s
  • \( V_{rel} = 150 \times \frac{5}{18} \text{ m/s} = \frac{750}{18} \text{ m/s} = \frac{125}{3} \text{ m/s} \)

Step 4: Calculate the Total Length for Crossing

For the two trains to completely cross each other, the total distance that needs to be covered at their relative speed is the sum of their individual lengths.

  • Total Length (\(L_{total}\)) = Length of Train A + Length of Train B
  • \( L_{total} = L_A + L_B = 400 \text{ m} + 500 \text{ m} = 900 \text{ m} \)

Step 5: Calculate the Time Taken to Cross Each Other

The time taken for the trains to cross each other is the total length required for crossing divided by their relative speed.

  • Time = \( \frac{\text{Total Length}}{\text{Relative Speed}} \)
  • Let \(T_{cross}\) be the time taken to cross.
  • \( T_{cross} = \frac{L_{total}}{V_{rel}} \)
  • \( T_{cross} = \frac{900 \text{ m}}{\frac{125}{3} \text{ m/s}} \)
  • \( T_{cross} = \frac{900 \times 3}{125} \text{ seconds} \)
  • \( T_{cross} = \frac{2700}{125} \text{ seconds} \)
  • \( T_{cross} = 21.6 \text{ seconds} \)

Therefore, the time taken by the two trains to cross each other is 21.6 seconds.

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Similar Questions

  1. Train \(X\) crosses a man standing on the platform in 24 seconds and train \(Y\) crosses a man standing on the platform in 18 seconds. They cross each other while running in opposite directions in 20 seconds. What is the ratio of speed of \(X\) to speed of \(Y\)?

  2. How far from station Q will the two trains meet?
     

  3. At what time will the two trains meet?


Important Questions from Problem on Trains

  1. Eight railway stations A, B, C, D, E, F, G and H are connected either by two-way passages or one-way passages. One-way passages are from C to A, E to G, B to F, D to H, G to C, E to C and H to G. Two-way passages are between A and E, G and B, F and D, and E and D.

    If the route between G and C is closed, which one of the following stations need not be passed through while travelling from H to C?

  2. A daily train is to be introduced between station A and station B starting from each end at 6 AM and the journey is to be completed in 42 hours. What is the number of trains needed in order to maintain the Shuttle Service?

  3. A train with a uniform speed passes a 122 meters long platform in 17 seconds and a 210 meters long bridge in 25 seconds. The speed of the train is:

  4. How long does a train 153 meters long running at the rate of 90 kmph take to cross a bridge 622 meters in length?

  5. A train passes a 360 metre long platform in 40 seconds and a man standing on the platform in 16 seconds. The speed of the train is:

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