Calculating the Time for Two Trains to Cross Each Other

Understanding the time it takes for two trains to cross each other is a fundamental concept in physics. This article explains the calculation process under different scenarios, including the standard problem and detailed analysis.

Standard Problem and Solution

The problem statement is as follows: Train A is 120m long and traveling at 20m/s. Train B is 130m long and traveling at 30 m/s. We need to determine the time it takes Train B to cross paths with Train A.

The total length of the two trains is the sum of their individual lengths:

Total distance 120m 130m 250m

The combined speed (since they are traveling towards each other) is:

Total speed 20m/s 30m/s 50m/s

Calculation

The time taken can be calculated using the formula: Time Total distance / Total speed

Plugging in the values:

Time 250m / 50m/s 5 seconds

Transformation into the First Train’s Rest Frame

In this scenario, we consider the first train (Train A) as stationary and observe the second train (Train B) moving at a relative speed of 50 m/s. The total distance that needs to be covered is:

Distance 120m 130m 250m

The time taken to cross each other under this rest frame is:

Time 250m / 50m/s 5 seconds

Detailed Analysis for Crossing Scenarios

In a more detailed analysis, we consider the different scenarios where crossing could occur:

Front of B to Rear of B

In this case, the rear of Train B crosses the rear of Train A, indicating a complete crossing.

The time calculation is as follows:

The front of B crosses the front of A at time 0. The front of B crosses the end of A at t 130m / 50m/s 2.6 seconds. The rear of B crosses the front of A at t 120m / 50m/s 2.4 seconds. The rear of B crosses the rear of A at t 2.4 seconds 130m / 50m/s 5 seconds.

So, if crossing means from the Front of B to rear of B, the answer is 2.6 seconds.

Fully Crossing Each Other

If crossing refers to the entire Train B passing the entire Train A, starting from the moment the fronts of each train cross, the answer is:

The time taken is 5 seconds.

Conclusion

The time it takes for Train B to cross Train A is typically estimated as 5 seconds when both trains are moving towards each other. However, the precise crossing time depends on the specific scenario and reference frame considered. This article provides a comprehensive explanation of the calculation methods and scenarios.