A high-speed train X takes 32 seconds to completely pass a signal post, and it requires 72 seconds to travel through a tunnel that is 1200 meters long. Another train Y, which has a length of 1280 meters, travels at a constant speed of 72 km/hr. If both trains are moving along parallel tracks in the same direction, calculate the total duration in seconds needed for train X to completely overtake train Y.

Correct Answer: 224 seconds

Step-by-Step Explanation:

Let us break down the problem into simple steps to find the time taken for train X to completely overtake train Y.

Step 1: Determine the speed and length of Train X

Let the length of train X be $L_X$ meters, and its speed be $S_X$ m/s.

When train X passes a signal post, the distance covered is equal to the length of the train itself.

Given, time taken to pass the signal post = 32 seconds.

Therefore, we can write the equation:

$L_X=S_X\times 32$

or $S_X=\frac{L_X}{32}$    --- (Equation 1)

When train X travels through a tunnel of length 1200 meters, the total distance covered is the sum of the length of the train and the length of the tunnel.

Total distance = $L_X+1200$ meters

Time taken to pass the tunnel = 72 seconds.

Using the speed formula:

$S_X=\frac{L_X+1200}{72}$    --- (Equation 2)

Equating Equation 1 and Equation 2:

$\frac{L_X}{32}=\frac{L_X+1200}{72}$

Cross-multiplying to solve for $L_X$:

$72\times L_X=32\times (L_X+1200)$

$72L_X=32L_X+38400$

$40L_X=38400$

$L_X=960\text{meters}$

Now, substitute $L_X=960$ back into Equation 1 to find the speed of train X ($S_X$):

$S_X=\frac{960}{32}=30\text{m/s}$

Step 2: Convert the speed of Train Y to m/s

Speed of train Y, $S_Y=72\text{km/hr}$

Converting km/hr to m/s by multiplying by $\frac{5}{18}$:

$S_Y=72\times \frac{5}{18}=20\text{m/s}$

Length of train Y, $L_Y=1280\text{meters}$

Step 3: Calculate the time required for Train X to overtake Train Y

Since both trains move in the same direction, the relative speed is the difference between their speeds:

Relative speed = $S_X-S_Y=30-20=10\text{m/s}$

The total distance to be covered for train X to completely overtake train Y is the sum of the lengths of both trains:

Total distance = $L_X+L_Y=960+1280=2240\text{meters}$

Time taken to overtake = $\frac{\text{Total Distance}}{\text{Relative Speed}}$

$\text{Time}=\frac{2240}{10}=224\text{seconds}$

Thus, the total duration needed for train X to completely overtake train Y is 224 seconds.