Six particles situated at the corners of a regular hexagon of side a move at a constant speed v. Each particle maintains a direction towards the particle at the next corner. Calculate the time the particles will take to meet each other.

Explanation

To solve the problem of six particles situated at the corners of a regular hexagon of side length \( a \), moving at a constant speed \( v \) and maintaining a direction towards the particle at the next corner, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Movement**:
Each particle moves towards the next particle in a circular path, forming a spiral inward towards the center of the hexagon.

2. **Analyzing the Geometry**:
The angle between the direction of motion of any two adjacent particles is \( 60^\circ \) because the internal angles of a regular hexagon are \( 120^\circ \), and the direction of motion forms a \( 60^\circ \) angle with the line connecting the two particles.

3. **Calculating the Component of Velocity**:
When one particle moves towards the next, we need to find the component of its velocity that is directed towards the next particle.
- The component of the velocity \( v \) in the direction towards the next particle is given by:
\[ v_{\text{towards}} = v \cos(60^\circ) = v \cdot \frac{1}{2} = \frac{v}{2} \]

4. **Relative Velocity**:
Since both particles are moving towards each other, the effective relative velocity at which they approach each other is:
\[ v_{\text{relative}} = v + v_{\text{towards}} = v + \frac{v}{2} = \frac{3v}{2} \]

5. **Distance to Travel**:
The distance each particle has to travel to meet the other is the side length of the hexagon, which is \( a \).

6. **Calculating Time to Meet**:
The time \( t \) taken for the particles to meet can be calculated using the formula:
\[ t = \frac{\text{Distance}}{\text{Relative Velocity}} = \frac{a}{\frac{3v}{2}} = \frac{2a}{3v} \]

### Final Answer:
The time taken for the particles to meet each other is:
\[ t = \frac{2a}{3v} \]