Solving Truck and Train Crossing Time Problem

This problem involves calculating the time it takes for a faster-moving object (truck) to completely pass a slower-moving object (train) when both are moving in the same direction.

Key Concepts: Relative Speed and Total Distance

• Relative Speed: When two objects move in the same direction, their relative speed is the difference between their individual speeds. This is the speed at which the faster object gains on the slower object.
• Total Distance: For the truck to completely cross the train, the effective distance to be covered is the sum of the length of the truck and the length of the train.

Given Information

• Length of Truck ($L_T$): 1575 metres
• Speed of Truck ($S_T$): 109 km/hr
• Length of Train ($L_{Tr}$): 1475 metres
• Speed of Train ($S_{Tr}$): 79 km/hr
• Direction: Same

Step-by-Step Calculation

1. Calculate Relative Speed:

   Since they are moving in the same direction, the relative speed ($S_{Rel}$) is:

   \( S_{Rel} = S_T - S_{Tr} \)

   \( S_{Rel} = 109 \text{ km/hr} - 79 \text{ km/hr} = 30 \text{ km/hr} \)

2. Convert Speed to Metres per Second (m/s):

   To find the time in seconds, convert the relative speed from km/hr to m/s using the conversion factor \(\frac{5}{18}\).

   \( S_{Rel} (\text{m/s}) = 30 \times \frac{5}{18} = \frac{150}{18} = \frac{25}{3} \text{ m/s} \)

3. Calculate Total Distance:

   The total distance ($D_{Total}$) the truck needs to cover relative to the train is the sum of their lengths.

   \( D_{Total} = L_T + L_{Tr} \)

   \( D_{Total} = 1575 \text{ m} + 1475 \text{ m} = 3050 \text{ m} \)

4. Calculate Time to Cross:

   Time ($T$) is calculated using the formula: Time = Total Distance / Relative Speed.

   \( T = \frac{D_{Total}}{S_{Rel} (\text{m/s})} \)

   \( T = \frac{3050 \text{ m}}{\frac{25}{3} \text{ m/s}} \)

   \( T = 3050 \times \frac{3}{25} \text{ s} \)

   \( T = 122 \times 3 \text{ s} \)

   \( T = 366 \text{ s} \)

Therefore, the truck will take 366 seconds to cross the train.