Four particles `A, B, C and D` are situated at the corner of a square `ABCD` of side a at t-0. Each of particles moves with constant speed (v). A always has its velocity along `AB, B` along `BC, C` along `CD~ and D` along `DA`. At what time will these particles meet each other ?

Answer

To solve the problem of when the four particles A, B, C, and D will meet at the center of the square ABCD, we can follow these steps:

### Step 1: Understand the motion of the particles
Each particle is moving with a constant speed \( v \) along the sides of the square. Particle A moves along side AB, B along BC, C along CD, and D along DA.

### Step 2: Identify the meeting point
The particles will meet at the center of the square, which we can denote as point O. The coordinates of point O can be determined as it is the midpoint of the diagonals of the square.

### Step 3: Calculate the distance from a corner to the center
The distance from any corner of the square (say A) to the center O can be calculated using the Pythagorean theorem. Since the side of the square is \( a \), the distance AO is:
\[ AO = \sqrt{\left(\frac{a}{2}\right)^2 + \left(\frac{a}{2}\right)^2} = \sqrt{\frac{a^2}{4} + \frac{a^2}{4}} = \sqrt{\frac{a^2}{2}} = \frac{a}{\sqrt{2}} \]

### Step 4: Determine the effective velocity towards the center
Since particle A is moving towards the center O at an angle of 45 degrees, we can find the component of its velocity in the direction of AO. The effective velocity component \( v_{AO} \) can be calculated as:
\[ v_{AO} = v \cos(45^\circ) = v \cdot \frac{1}{\sqrt{2}} = \frac{v}{\sqrt{2}} \]

### Step 5: Calculate the time taken to meet at the center
Using the formula for time, \( t = \frac{\text{distance}}{\text{velocity}} \), we can find the time taken by particle A to reach point O:
\[ t = \frac{AO}{v_{AO}} = \frac{\frac{a}{\sqrt{2}}}{\frac{v}{\sqrt{2}}} = \frac{a}{v} \]

### Conclusion
Thus, the time at which all four particles will meet at the center O is:
\[ t = \frac{a}{v} \]