six particles are situated at the corners of a regular hexagon. These particles start moving with equal speed of 10 m/s in such that velocity of any one particle is directed towards the next particle. Find the rate (in m/s) at which length of a side of the hexagon is decreasing.

Answer

To solve the problem of determining the rate at which the length of a side of the hexagon is decreasing, we can follow these steps:

### Step 1: Understand the Configuration
We have six particles located at the corners of a regular hexagon. Each particle moves towards the next particle with a speed of 10 m/s. Let's label the particles as A, B, C, D, E, and F in a clockwise manner.

**Hint:** Visualize the hexagon and the direction of motion of each particle towards the next.

### Step 2: Analyze the Motion
Each particle moves directly towards the next one. For example, particle A moves towards particle B, particle B moves towards particle C, and so on. The angle between the direction of motion of any particle and the line connecting it to the next particle is 60 degrees (since the internal angles of a regular hexagon are 120 degrees).

**Hint:** Remember that the angle between the velocity vector of a particle and the line connecting it to the next particle is crucial for resolving the velocities.

### Step 3: Resolve the Velocity Components
The velocity of each particle can be resolved into two components: one along the line connecting it to the next particle and one perpendicular to that line.

For particle A moving towards B:
- The component of the velocity of A towards B is \( v_{AB} = v \cos(60^\circ) = 10 \cdot \frac{1}{2} = 5 \, \text{m/s} \).

**Hint:** Use trigonometric functions to resolve the velocities into components along the direction of interest.

### Step 4: Determine the Rate of Decrease of the Side Length
Since particle A is moving towards particle B with a velocity of 5 m/s, this means that the distance between particles A and B (which is the length of the side of the hexagon) is decreasing at this rate.

Thus, the rate at which the length of a side of the hexagon is decreasing is \( 5 \, \text{m/s} \).

**Hint:** The rate of decrease in distance between two moving particles can be directly derived from the component of their velocities along the line connecting them.

### Final Answer
The rate at which the length of a side of the hexagon is decreasing is \( 5 \, \text{m/s} \).

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