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 \( v \) towards each other, we can break down the motion and calculate the time it takes for them to meet at the center. Here’s a step-by-step solution:

### Step 1: Understanding the Setup
We have six particles located at the corners of a regular hexagon. Let's label the particles as \( A, B, C, D, E, \) and \( F \). Each particle moves towards the next particle in a clockwise manner.

### Step 2: Analyzing the Motion
When particle \( A \) moves towards particle \( B \), it is not moving directly towards the center of the hexagon, but rather at an angle. The same applies to all other particles.

### Step 3: Determine the Angle
In a regular hexagon, the angle subtended at the center by each side is \( 60^\circ \). Therefore, the angle \( \theta \) between the direction of motion of each particle and the line connecting the center of the hexagon to the particle is \( 30^\circ \).

### Step 4: Velocity Components
The velocity of each particle can be resolved into two components:
- The component towards the center of the hexagon, which is \( v \cos(30^\circ) \).
- The component perpendicular to the line connecting the particles, which does not contribute to closing the distance between them.

### Step 5: Calculate the Closing Speed
Since each particle is moving towards the next one, the effective closing speed between any two adjacent particles (say \( A \) and \( B \)) can be calculated as:
\[ v_{\text{effective}} = v - v \cos(30^\circ) \]
Using \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \), we find:
\[ v_{\text{effective}} = v - v \cdot \frac{\sqrt{3}}{2} = v \left(1 - \frac{\sqrt{3}}{2}\right) \]

### Step 6: Calculate the Time to Meet
The initial distance between any two adjacent particles is \( a \). The time \( t \) taken for the particles to meet can be calculated using the formula:
\[ t = \frac{\text{Distance}}{\text{Relative Velocity}} = \frac{a}{v_{\text{effective}}} \]
Substituting the effective velocity:
\[ t = \frac{a}{v \left(1 - \frac{\sqrt{3}}{2}\right)} \]

### Step 7: Simplifying the Expression
To simplify the expression, we can calculate \( 1 - \frac{\sqrt{3}}{2} \) and substitute it back into the equation. However, for practical purposes, we can keep it in this form.

### Final Answer
Thus, the time taken for the particles to meet at the center of the hexagon is:
\[ t = \frac{a}{v \left(1 - \frac{\sqrt{3}}{2}\right)} \]