Four particles situated at the corners of a square of side ‘a’ move at a constant speed v. Each particle maintains a direction towards the next particle in succession. Calculate the time particles will take to meet each other.

Options

A. `a/v`
B. `a/(2v)`
C. `(a)/(3v)`
D. `(2a)/(3v)`

Answer

To solve the problem of four particles situated at the corners of a square of side 'a' moving towards each other, we can follow these steps:

### Step 1: Understand the Configuration
The four particles are located at the corners of a square. Let’s label the corners as A, B, C, and D. Each particle moves towards the next particle in a clockwise direction:
- Particle A moves towards Particle B
- Particle B moves towards Particle C
- Particle C moves towards Particle D
- Particle D moves towards Particle A

### Step 2: Analyze the Motion
Since each particle is always moving towards the next particle, they will spiral inward towards the center of the square. The distance between any two adjacent particles (e.g., A and B) is initially 'a'.

### Step 3: Determine the Relative Velocity
To find out how long it will take for the particles to meet, we need to consider the relative velocity of one particle towards another. For example, consider Particle A moving towards Particle B. The speed of Particle A is 'v', and since it is directed towards Particle B, we need to find the component of this velocity in the direction of the line joining A and B.

### Step 4: Calculate the Time to Meet
1. The distance between A and B is 'a'.
2. The relative velocity of A towards B is 'v' because Particle A is moving directly towards Particle B.
3. The time taken \( t \) for Particle A to meet Particle B can be calculated using the formula:
\[ t = \frac{\text{Distance}}{\text{Relative Velocity}} = \frac{a}{v} \]

### Step 5: Conclusion
Since all particles are moving towards each other in a similar manner, they will all meet at the same time. Therefore, the time taken for all four particles to meet at the center of the square is:
\[ t = \frac{a}{v} \]

### Final Answer
The time taken for the particles to meet each other is \( \frac{a}{v} \).

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