Two trains travel in the same direction at 50 km/hr and 32 km/hr respectively. A man in the slower train observes that the faster train passes him completely in 15 seconds. What is the length of the faster train in metres?

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Determine the Speeds of the Trains
The speeds of the two trains are given as:
- Speed of the faster train = 50 km/hr
- Speed of the slower train = 32 km/hr

### Step 2: Calculate the Relative Speed
Since both trains are moving in the same direction, the relative speed of the faster train with respect to the slower train can be calculated by subtracting the speed of the slower train from the speed of the faster train.

\[ \text{Relative Speed} = \text{Speed of Faster Train} - \text{Speed of Slower Train} \]
\[ \text{Relative Speed} = 50 \text{ km/hr} - 32 \text{ km/hr} = 18 \text{ km/hr} \]

### Step 3: Convert Relative Speed to Meters per Second
To find the length of the faster train in meters, we need to convert the relative speed from kilometers per hour to meters per second. The conversion factor is:
\[ 1 \text{ km/hr} = \frac{1000 \text{ meters}}{3600 \text{ seconds}} = \frac{5}{18} \text{ m/s} \]

Now, we convert 18 km/hr to m/s:
\[ \text{Relative Speed in m/s} = 18 \times \frac{5}{18} = 5 \text{ m/s} \]

### Step 4: Use the Time Taken to Pass the Man
The problem states that the faster train passes the man in the slower train completely in 15 seconds.

### Step 5: Calculate the Length of the Faster Train
The length of the faster train can be calculated using the formula:
\[ \text{Length} = \text{Speed} \times \text{Time} \]
Substituting the values we have:
\[ \text{Length of Faster Train} = 5 \text{ m/s} \times 15 \text{ seconds} = 75 \text{ meters} \]

### Final Answer
The length of the faster train is **75 meters**.

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