A 400 m long train takes 36 s to cross a man walking at 20km/h in the direction opposite to that of the train. What is the speed of the train?

Options

A. 20 km/h
B. 30 km/h
C. 15 km/h
D. 11 km/h

Answer

To solve the problem step by step, we will follow the given information and apply the relevant formulas.

### Step 1: Understand the given data
- Length of the train (L) = 400 meters
- Time taken to cross the man (T) = 36 seconds
- Speed of the man (Vm) = 20 km/h

### Step 2: Convert the speed of the man from km/h to m/s
To convert km/h to m/s, we use the conversion factor \( \frac{5}{18} \):
\[ Vm = 20 \text{ km/h} \times \frac{5}{18} = \frac{100}{18} \approx 5.56 \text{ m/s} \]

### Step 3: Calculate the relative speed of the train with respect to the man
Let the speed of the train be \( Vt \) km/h. Since the train and the man are moving in opposite directions, the relative speed (Vr) is:
\[ Vr = Vt + Vm \]
In m/s, this becomes:
\[ Vr = \left(Vt \times \frac{5}{18}\right) + 5.56 \]

### Step 4: Use the formula for distance
The distance covered by the train while crossing the man is equal to the length of the train:
\[ \text{Distance} = \text{Relative Speed} \times \text{Time} \]
Substituting the known values:
\[ 400 = \left(Vt \times \frac{5}{18} + 5.56\right) \times 36 \]

### Step 5: Simplify the equation
First, simplify the right side:
\[ 400 = \left(Vt \times \frac{5}{18} + 5.56\right) \times 36 \]
\[ 400 = Vt \times \frac{5 \times 36}{18} + 5.56 \times 36 \]
\[ 400 = Vt \times 10 + 200.16 \]

### Step 6: Solve for \( Vt \)
Rearranging the equation gives:
\[ 400 - 200.16 = Vt \times 10 \]
\[ 199.84 = Vt \times 10 \]
\[ Vt = \frac{199.84}{10} \approx 19.984 \text{ km/h} \]

### Step 7: Round off the speed of the train
Rounding off gives:
\[ Vt \approx 20 \text{ km/h} \]

### Final Answer
The speed of the train is approximately **20 km/h**.

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