n' number of particles are located at the verticles of a regular polygon of n sides having the edge length 'a'. They all start moving simultaneously with equal constant speed 'v' heading towards each other all the time. How long will the particles take to collide?

Explanation

To solve the problem of how long it will take for n particles located at the vertices of a regular polygon with n sides and edge length 'a' to collide, we can follow these steps:

### Step 1: Understand the Geometry of the Problem
The particles are located at the vertices of a regular polygon. Each particle moves towards the next particle, and the angle between adjacent sides of the polygon is given by \( \theta = \frac{2\pi}{n} \).

**Hint:** Visualize the polygon and the direction of motion of the particles.

### Step 2: Determine the Relative Velocity
Each particle moves with a constant speed \( v \). When a particle at vertex A moves towards the next particle at vertex B, the effective component of its velocity towards B can be calculated using the cosine of the angle \( \theta \):
- The velocity of particle A towards B is \( v \).
- The velocity of particle B towards A is \( v \cos(\theta) \).

The relative velocity \( V_{AB} \) of particle A with respect to particle B can be expressed as:
\[ V_{AB} = v + v \cos(\theta) = v(1 + \cos(\theta)) \]

**Hint:** Remember that the relative velocity accounts for the direction of motion of both particles.

### Step 3: Substitute the Angle
Substituting the expression for \( \theta \):
\[ \theta = \frac{2\pi}{n} \]
We can rewrite the relative velocity as:
\[ V_{AB} = v \left(1 + \cos\left(\frac{2\pi}{n}\right)\right) \]

**Hint:** Use the cosine of the angle to find the effective velocity towards each other.

### Step 4: Calculate the Time to Collide
The time \( t \) taken for the particles to collide can be calculated using the formula:
\[ t = \frac{\text{Distance}}{\text{Relative Velocity}} \]
The distance each particle needs to travel to collide is the length of one side of the polygon, which is \( a \). Thus, we have:
\[ t = \frac{a}{V_{AB}} = \frac{a}{v \left(1 + \cos\left(\frac{2\pi}{n}\right)\right)} \]

**Hint:** Ensure that you are using the correct distance and relative velocity in your calculations.

### Final Answer
The time taken for the particles to collide is:
\[ t = \frac{a}{v \left(1 + \cos\left(\frac{2\pi}{n}\right)\right)} \]