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 velocity 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 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. The distance the train needs to cover to completely overtake the man is equal to its length, which is 200 meters.

Using the formula:
\[ \text{Relative Speed} = \frac{\text{Distance}}{\text{Time}} \]
Here, the distance is 200 meters and the time is 20 seconds.

So, we have:
\[ \text{Relative Speed} = \frac{200 \text{ m}}{20 \text{ s}} = 10 \text{ m/s} \]

Since the train is moving at 20 m/s and the relative speed is the difference in their speeds when moving in the same direction:
\[ 20 - V_m = 10 \]
Where \( V_m \) is the speed of the man.

Rearranging gives:
\[ V_m = 20 - 10 = 10 \text{ m/s} \]

### Step 3: Calculate the time taken when the man runs in the opposite direction
When the man runs in the opposite direction, the relative speed will be the sum of the speeds of the train and the man:
\[ \text{Relative Speed} = V_t + V_m = 20 + 10 = 30 \text{ m/s} \]

Now, we will use the same formula for time:
\[ \text{Time} = \frac{\text{Distance}}{\text{Relative Speed}} \]
The distance to be covered is still 200 meters.

So, we have:
\[ \text{Time} = \frac{200 \text{ m}}{30 \text{ m/s}} = \frac{200}{30} = \frac{20}{3} \text{ seconds} \]

### Final Answer
The train will take \( \frac{20}{3} \) seconds to overtake the man if he is running in the opposite direction.

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