A train 800 metres long is running at a speed of 78 km/h. It passes completely through a tunnel in 1 minute. What is the length of the tunnel in metres?

Introduction / Context:
This tunnel problem is a direct application of uniform motion. It tests whether you can compute the total distance a train travels while it completely passes through a tunnel and then subtract the train length to obtain the tunnel length. It also requires correct conversion from km/h to metres per second and accurate use of the time in seconds.

Given Data / Assumptions:

• Length of the train = 800 metres.
• Speed of the train = 78 km/h.
• Time taken to completely pass through the tunnel = 1 minute = 60 seconds.
• The train travels at a constant speed.
• We must find the length of the tunnel.

Concept / Approach:
To clear the tunnel completely, the train must travel a distance equal to the sum of its own length and the tunnel length. We first convert the speed from km/h to metres per second. Then, using distance = speed * time, we find the total distance covered while inside the tunnel. Finally, subtracting the train length from this total distance yields the tunnel length.

Step-by-Step Solution:
Step 1: Convert speed from km/h to metres per second: 78 km/h = 78 * 5 / 18 metres per second.Step 2: Compute 78 * 5 / 18 = 390 / 18 = 65 / 3 metres per second.Step 3: Time taken to pass the tunnel completely = 1 minute = 60 seconds.Step 4: Total distance travelled while passing the tunnel = speed * time = (65 / 3) * 60 = 65 * 20 = 1300 metres.Step 5: Let tunnel length be T metres. Then total distance = train length + tunnel length = 800 + T.Step 6: So 800 + T = 1300, giving T = 1300 - 800 = 500 metres.

Verification / Alternative check:
If the tunnel length is 500 metres, total distance to clear it is 800 + 500 = 1300 metres.At 65 / 3 metres per second, time = 1300 / (65 / 3) = 1300 * 3 / 65 = 60 seconds, which is exactly 1 minute as given in the question.

Why Other Options Are Wrong:
Tunnel lengths like 130 metres or 360 metres are too short and would result in a shorter passing time than 60 seconds. Values such as 420 metres or 540 metres do not lead to the exact 60 second figure when recomputed. Only 500 metres produces the correct time with the given speed and train length.

Common Pitfalls:
Candidates sometimes forget to add the train length to the tunnel length when computing total distance, using only the tunnel length instead. Another frequent mistake is incorrect unit conversion from km/h to metres per second, or misinterpreting 1 minute as 1 hour. Consistent units and a clear understanding of the geometry of the situation are key to solving this type of question.

Final Answer:
The length of the tunnel is 500 metres.