A train 900 metres long is running at a speed of 78 km/h. If it crosses a tunnel completely in 1 minute, what is the length of the tunnel in metres?

Introduction / Context:
This train problem focuses on calculating the length of a tunnel when a train of known length passes through it in a given time. To clear the tunnel completely, the train must travel a distance equal to its own length plus the length of the tunnel. With the speed of the train and the total time provided, we can determine the tunnel length.

Given Data / Assumptions:

• Length of the train = 900 m.
• Speed of the train = 78 km/h.
• Time to cross the tunnel completely = 1 minute = 60 s.
• Speed is uniform and track is straight.
• Distance covered during the 60 s equals train length plus tunnel length.

Concept / Approach:
We convert the train speed from km/h to m/s to match the distance in metres. Then we calculate the total distance travelled in 60 seconds. From this total distance, we subtract the known train length to obtain the tunnel length. The key formula used is distance = speed * time.

Step-by-Step Solution:
Step 1: Convert speed from km/h to m/s. Step 2: 78 km/h = 78 * 5/18 = 390/18 = 21.666... m/s. Step 3: Total distance covered in 60 seconds = 21.666... * 60 ≈ 1300 m. Step 4: This total distance equals train length + tunnel length. Step 5: Tunnel length = 1300 - 900 = 400 m. Step 6: Hence, the tunnel is 400 metres long.

Verification / Alternative check:
We can recalculate more exactly using fractions. 78 km/h = 78 * 5/18 = 390/18 = 65/3 m/s. In 60 s, distance = (65/3) * 60 = 65 * 20 = 1300 m. Subtracting the train length 900 m gives 400 m exactly. The arithmetic confirms the tunnel length without relying on decimal approximations.

Why Other Options Are Wrong:
A tunnel of 300 m or 350 m would result in a total distance of 1200 or 1250 m respectively, which does not match the calculated 1300 m. Options 500 m and 600 m give total distances of 1400 and 1500 m, which are too large for the same speed and time combination.

Common Pitfalls:
Learners sometimes mistakenly use only the tunnel length as the distance for complete crossing, forgetting that the train must also move its full length. Another pitfall is incorrect conversion from km/h to m/s or using 60 minutes instead of 60 seconds. Keeping track of units throughout the solution is crucial.

Final Answer:
The length of the tunnel is 400 m.