Two trains pass each other on parallel lines. Each train is 100 m long. When they are going in the same direction, the faster one takes 60 seconds to pass the other completely. If they are going in opposite directions, they pass each other completely in 10 seconds. Find the speed of the slower train in km/ hr.

Options

A. 30 km/hr
B. 42 km/hr
C. 48 km/hr
D. 60 km/hr

Answer

To solve the problem step by step, we will use the information provided about the two trains passing each other in different directions and apply the concepts of relative speed.

### Step 1: Understand the Problem
We have two trains, each 100 meters long. When they are moving in the same direction, the faster train takes 60 seconds to completely pass the slower train. When they are moving in opposite directions, they take 10 seconds to pass each other.

### Step 2: Define Variables
Let:
- Speed of the faster train = \( x \) km/hr
- Speed of the slower train = \( y \) km/hr

### Step 3: Convert Speeds to m/s
To work with meters and seconds, we convert the speeds from km/hr to m/s using the conversion factor \( \frac{5}{18} \):
- Speed of the faster train in m/s = \( \frac{5}{18} x \)
- Speed of the slower train in m/s = \( \frac{5}{18} y \)

### Step 4: Set Up Equations for Same Direction
When the trains are moving in the same direction, the relative speed is:
\[ \text{Relative Speed} = \left( \frac{5}{18} x - \frac{5}{18} y \right) \]
The total distance to be covered when passing each other is the sum of their lengths:
\[ \text{Distance} = 100 + 100 = 200 \text{ m} \]
Using the formula:
\[ \text{Distance} = \text{Relative Speed} \times \text{Time} \]
We have:
\[ 200 = \left( \frac{5}{18} (x - y) \right) \times 60 \]
Simplifying this gives:
\[ 200 = \frac{5}{18} (x - y) \times 60 \]
\[ 200 = \frac{5 \times 60}{18} (x - y) \]
\[ 200 = \frac{300}{18} (x - y) \]
\[ 200 = \frac{50}{3} (x - y) \]
Multiplying both sides by 3:
\[ 600 = 50 (x - y) \]
Dividing by 50:
\[ x - y = 12 \quad \text{(Equation 1)} \]

### Step 5: Set Up Equations for Opposite Direction
When the trains are moving in opposite directions, the relative speed is:
\[ \text{Relative Speed} = \left( \frac{5}{18} x + \frac{5}{18} y \right) \]
Using the same distance:
\[ 200 = \left( \frac{5}{18} (x + y) \right) \times 10 \]
Simplifying this gives:
\[ 200 = \frac{5}{18} (x + y) \times 10 \]
\[ 200 = \frac{50}{18} (x + y) \]
Multiplying both sides by 18:
\[ 3600 = 50 (x + y) \]
Dividing by 50:
\[ x + y = 72 \quad \text{(Equation 2)} \]

### Step 6: Solve the Equations
Now we have a system of equations:
1. \( x - y = 12 \)
2. \( x + y = 72 \)

Adding these two equations:
\[ (x - y) + (x + y) = 12 + 72 \]
\[ 2x = 84 \]
\[ x = 42 \text{ km/hr} \] (Speed of the faster train)

Now substitute \( x \) back into one of the equations to find \( y \):
Using Equation 2:
\[ 42 + y = 72 \]
\[ y = 72 - 42 \]
\[ y = 30 \text{ km/hr} \] (Speed of the slower train)

### Final Answer
The speed of the slower train is **30 km/hr**.