Answer :
To solve this problem, we need to find the length of the bridge that a train passes over in 35 seconds while traveling at 72 km/h. Additionally, we are given that the train takes 1 minute 52 seconds to overtake another train of the same length traveling at 54 km/h.
Step 1: Convert Speeds to Meters per Second
To work with consistent units, we first convert the speeds from kilometers per hour to meters per second.
Speed of First Train:
[tex]72 \text{ km/h} = \frac{72 \times 1000}{3600} \text{ m/s} = 20 \text{ m/s}[/tex]Speed of Second Train:
[tex]54 \text{ km/h} = \frac{54 \times 1000}{3600} \text{ m/s} = 15 \text{ m/s}[/tex]
Step 2: Find Length of the Train
When the first train crosses the bridge in 35 seconds, the distance it covers (which is the combined length of the train and the bridge) can be calculated using its speed:
[tex]\text{Distance covered in crossing the bridge} = \text{Speed} \times \text{Time} = 20 \times 35 = 700 \text{ meters}[/tex]
Step 3: Calculate the Relative Speed and Time to Overtake
The first train overtakes the second train, and it takes 1 minute 52 seconds to do so. First, convert the time to seconds:
[tex]1 \text{ minute} 52 \text{ seconds} = 60 + 52 = 112 \text{ seconds}[/tex]
The relative speed between the two trains is:
[tex]\text{Relative Speed} = (20 - 15) \text{ m/s} = 5 \text{ m/s}[/tex]
In this time, the first train covers a distance equal to the length of both trains:
[tex]2 \times \text{Train Length} = 5 \times 112 = 560 \text{ meters}[/tex]
This means:
[tex]\text{Length of One Train} = \frac{560}{2} = 280 \text{ meters}[/tex]
Step 4: Calculate the Length of the Bridge
We have already calculated the total distance the train covers while passing the bridge as 700 meters. Since one train length is 280 meters:
[tex]\text{Length of the Bridge} = 700 - 280 = 420 \text{ meters}[/tex]
Conclusion
The length of the bridge is [tex]420[/tex] meters.
Thus, the correct option is (4) 420 m.
