A 200 m long train moving with velocity `20 ms^(-1)` overtakes a man running in the same direction in 20 seconds. How long the train will take to overtake the man if he was running in opposite direction?

Options

A. `10/3 s`
B. `20/3 s`
C. `5/3 s`
D. 3s

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the Problem
We have a train that is 200 meters long, moving at a speed of 20 m/s. It overtakes a man running in the same direction in 20 seconds. We need to find out how long it will take for the train to overtake the same man if he is running in the opposite direction.

### Step 2: Calculate the Speed of the Man
When the train overtakes the man in the same direction, we can use the formula for relative speed.

Let the speed of the man be \( V \) m/s. The relative speed of the train with respect to the man is:
\[ \text{Relative Speed} = \text{Speed of Train} - \text{Speed of Man} = 20 - V \text{ m/s} \]

The time taken to overtake the man is given as 20 seconds. The distance covered by the train while overtaking the man is equal to the length of the train, which is 200 meters.

Using the formula:
\[ \text{Distance} = \text{Relative Speed} \times \text{Time} \]
we can write:
\[ 200 = (20 - V) \times 20 \]

### Step 3: Solve for V
Now, let's solve for \( V \):
\[ 200 = 20 \times (20 - V) \]
\[ 200 = 400 - 20V \]
\[ 20V = 400 - 200 \]
\[ 20V = 200 \]
\[ V = \frac{200}{20} = 10 \text{ m/s} \]

### Step 4: Calculate the Time to Overtake When Running in Opposite Direction
Now, if the man is running in the opposite direction, the relative speed of the train with respect to the man will be:
\[ \text{Relative Speed} = \text{Speed of Train} + \text{Speed of Man} = 20 + 10 = 30 \text{ m/s} \]

Using the same distance formula, we can find the time taken to overtake:
\[ \text{Distance} = \text{Relative Speed} \times \text{Time} \]
\[ 200 = 30 \times \text{Time} \]
\[ \text{Time} = \frac{200}{30} = \frac{20}{3} \text{ seconds} \]

### Final Answer
The time taken for the train to overtake the man when he is running in the opposite direction is \( \frac{20}{3} \) seconds.

---