Four persons A, B, C and D initially at the corners of a square of side of length d. If every person starts moving with same speed v such that each one faces the other always, the person will meet after time

Options

A. `d/v`
B. `(sqrt(2)d)/(v)`
C. `(d)/(2v)`
D. `(d)/(sqrt(2)v)`

Answer

To solve the problem of four persons A, B, C, and D starting at the corners of a square and moving towards each other while always facing one another, we can follow these steps:

### Step 1: Understand the Configuration
We have a square with side length \( d \). The four persons are positioned at the corners of the square:
- A at (0, 0)
- B at (d, 0)
- C at (d, d)
- D at (0, d)

### Step 2: Determine the Movement Direction
Each person moves towards the person they are facing. Since they are always facing each other, they will move in a curved path towards the center of the square.

### Step 3: Calculate the Distance to the Center
The center of the square can be found at the coordinates:
\[ \left(\frac{d}{2}, \frac{d}{2}\right) \]
The distance from any corner (say A) to the center is given by:
\[ \text{Distance} = \sqrt{\left(\frac{d}{2} - 0\right)^2 + \left(\frac{d}{2} - 0\right)^2} = \sqrt{\left(\frac{d}{2}\right)^2 + \left(\frac{d}{2}\right)^2} = \sqrt{\frac{d^2}{4} + \frac{d^2}{4}} = \sqrt{\frac{d^2}{2}} = \frac{d}{\sqrt{2}} \]

### Step 4: Relate Distance, Speed, and Time
Each person moves with a constant speed \( v \). The time \( T \) taken to reach the center can be calculated using the formula:
\[ T = \frac{\text{Distance}}{\text{Speed}} = \frac{\frac{d}{\sqrt{2}}}{v} = \frac{d}{v\sqrt{2}} \]

### Conclusion
Thus, the time taken for all four persons to meet at the center of the square is:
\[ T = \frac{d}{v\sqrt{2}} \]