Calculating Train Length and Crossing Difference

This problem involves calculating the lengths of two trains and the difference between them based on their speeds and crossing times. We need to find the length of the first train using the time it takes to cross a stationary post, and then use the relative speed and crossing time to find the sum of their lengths. Finally, we can determine the length of the second train and the difference.

Step 1: Determine the Length of the First Train

The first train travels at 72 km/hr and crosses a post in 20 seconds. To find the length of the train, we first convert its speed to meters per second (m/s).

• Speed of the first train ($v_1$): 72 km/hr
• Conversion factor: $1 \text{ km/hr} = \frac{5}{18} \text{ m/s}$
• Speed in m/s: $v_1 = 72 \times \frac{5}{18} \text{ m/s} = 4 \times 5 \text{ m/s} = 20 \text{ m/s}$

When a train crosses a post, the distance covered is equal to the length of the train ($L_1$).

• Time to cross the post ($t_1$): 20 seconds
• Length of the first train ($L_1$) = Speed ($v_1$) $\times$ Time ($t_1$)
• $L_1 = 20 \text{ m/s} \times 20 \text{ s} = 400 \text{ m}$

So, the length of the first train is 400 meters.

Step 2: Calculate the Relative Speed

The first train (72 km/hr) crosses the second train (54 km/hr) while travelling in the same direction. The relative speed ($v_{rel}$) is the difference between their speeds.

• Speed of the first train ($v_1$): 72 km/hr
• Speed of the second train ($v_2$): 54 km/hr
• Relative speed ($v_{rel}$) = $v_1 - v_2$ (since they are moving in the same direction)
• $v_{rel} = 72 \text{ km/hr} - 54 \text{ km/hr} = 18 \text{ km/hr}$

Now, convert the relative speed to meters per second (m/s).

• $v_{rel} = 18 \times \frac{5}{18} \text{ m/s} = 5 \text{ m/s}$

Alternatively, we can convert both speeds to m/s first:

• $v_1 = 20 \text{ m/s}$
• $v_2 = 54 \times \frac{5}{18} \text{ m/s} = 3 \times 5 \text{ m/s} = 15 \text{ m/s}$
• $v_{rel} = v_1 - v_2 = 20 \text{ m/s} - 15 \text{ m/s} = 5 \text{ m/s}$

Step 3: Calculate the Sum of the Train Lengths

The time taken for the first train to cross the second train is 1 minute 45 seconds. When one train crosses another, the total distance covered relative to each other is the sum of their lengths ($L_1 + L_2$).

• Time to cross each other ($t_{cross}$): 1 minute 45 seconds
• Convert time to seconds: $t_{cross} = (1 \times 60) + 45 = 60 + 45 = 105 \text{ seconds}$
• Sum of lengths ($L_1 + L_2$) = Relative Speed ($v_{rel}$) $\times$ Time ($t_{cross}$)
• $L_1 + L_2 = 5 \text{ m/s} \times 105 \text{ s} = 525 \text{ m}$

The combined length of the two trains is 525 meters.

Step 4: Determine the Length of the Second Train

We know the length of the first train ($L_1$) and the sum of the lengths ($L_1 + L_2$). We can now find the length of the second train ($L_2$).

• $L_1 = 400 \text{ m}$
• $L_1 + L_2 = 525 \text{ m}$
• Length of the second train ($L_2$) = $(L_1 + L_2) - L_1$
• $L_2 = 525 \text{ m} - 400 \text{ m} = 125 \text{ m}$

The length of the second train is 125 meters.

Step 5: Calculate the Difference in Length

Finally, we calculate the difference between the lengths of the two trains.

• Difference = $|L_1 - L_2|$
• Difference = $|400 \text{ m} - 125 \text{ m}|$
• Difference = $275 \text{ m}$

The difference in length between the two trains is 275 meters.