A 500 m long train running at a speed of 100 m/s crosses a tunnel in 10 s. What is the length of the tunnel?

The correct answer is 500 m.

To solve this problem, we need to understand what happens when a train crosses a tunnel. When a train completely passes through a tunnel, the total distance covered by the train is equal to the sum of the length of the train and the length of the tunnel. This is because the front of the train must travel the full length of the tunnel, and then the tail of the train must also exit the tunnel completely.

Given Information:
Length of the train = 500 m
Speed of the train = 100 m/s
Time taken to cross the tunnel = 10 s

Step 1: Calculate the total distance covered.

Using the basic formula of motion:

$$\text{Distance}=\text{Speed}\times \text{Time}$$

$$\text{Distance}=100\text{m/s}\times 10\text{s}=1000\text{m}$$

So, the train travels a total distance of 1000 m during the crossing.

Step 2: Set up the relationship between the distances.

As explained above, the total distance covered equals the sum of both lengths:

$$\text{Total Distance}=\text{Length of Train}+\text{Length of Tunnel}$$

$$1000=500+\text{Length of Tunnel}$$

Step 3: Solve for the length of the tunnel.

$$\text{Length of Tunnel}=1000-500=500\text{m}$$

Conclusion: The length of the tunnel is 500 m. The key concept here is that the train must travel a distance equal to its own length plus the tunnel's length to fully cross it — the journey begins when the engine enters the tunnel and ends only when the last carriage exits.