A man is standing on a railway bridge which is 50 m long. He finds that a train crossed the bridge in `4(1)/(2)` sec but himself in 2sec. Find the length & speed of train.

Answer

To solve the problem step by step, we will follow the reasoning outlined in the video transcript.

### Step 1: Understand the scenario
A man is standing on a bridge that is 50 meters long. A train crosses the bridge in 4.5 seconds and the man in 2 seconds. We need to find the length and speed of the train.

### Step 2: Convert time into a fraction
The time taken by the train to cross the bridge is given as 4.5 seconds. We can convert this into an improper fraction:
\[ 4.5 = \frac{9}{2} \text{ seconds} \]

### Step 3: Set up the equation for the train crossing the bridge
When the train crosses the bridge, it covers the length of the bridge plus its own length. Let \( L \) be the length of the train. The total distance covered by the train is:
\[ \text{Distance} = \text{Length of the bridge} + \text{Length of the train} = 50 + L \]
The time taken to cross this distance is \( \frac{9}{2} \) seconds. Therefore, we can write:
\[ \text{Speed of the train} = \frac{50 + L}{\frac{9}{2}} \]

### Step 4: Set up the equation for the man crossing the train
When the train crosses the man, it only covers its own length \( L \) in 2 seconds. Thus, we can write:
\[ \text{Speed of the train} = \frac{L}{2} \]

### Step 5: Equate the two expressions for speed
Since both expressions represent the speed of the train, we can set them equal to each other:
\[ \frac{50 + L}{\frac{9}{2}} = \frac{L}{2} \]

### Step 6: Solve for \( L \)
Cross-multiplying gives:
\[ 2(50 + L) = 9L \]
Expanding and rearranging the equation:
\[ 100 + 2L = 9L \]
\[ 100 = 9L - 2L \]
\[ 100 = 7L \]
Dividing both sides by 7:
\[ L = \frac{100}{7} \approx 14.29 \text{ meters} \]

### Step 7: Calculate the speed of the train
Now, we can substitute \( L \) back into either speed equation. Using the equation \( \text{Speed} = \frac{L}{2} \):
\[ \text{Speed} = \frac{\frac{100}{7}}{2} = \frac{100}{14} = \frac{50}{7} \text{ m/s} \approx 7.14 \text{ m/s} \]

### Step 8: Convert speed to km/h
To convert the speed from meters per second to kilometers per hour, we multiply by \( \frac{18}{5} \):
\[ \text{Speed in km/h} = \frac{50}{7} \times \frac{18}{5} = \frac{180}{7} \approx 25.71 \text{ km/h} \]

### Final Answer
- Length of the train: \( \frac{100}{7} \approx 14.29 \) meters
- Speed of the train: \( \frac{50}{7} \approx 7.14 \) m/s or \( \approx 25.71 \) km/h