A particle moves with constant speed `v` along a regular hexagon `ABCDEF` in the same order. Then the magnitude of the avergae velocity for its motion form `A` to

Options

A. `F` is `v//5`
B. `D` is `v//3`
C. `C` is `vsqrt(3)//2`
D. `B` is `v`

Explanation

To solve the problem of finding the average velocity of a particle moving along a regular hexagon from point A to point F, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Geometry of the Hexagon**:
- A regular hexagon has six equal sides. Let the length of each side be `x`.
- The vertices of the hexagon are labeled as A, B, C, D, E, and F.

2. **Determine the Displacement from A to F**:
- The displacement from point A to point F can be visualized as a straight line connecting these two points.
- Since A and F are opposite vertices of the hexagon, the displacement is equal to the length of the line segment connecting A and F.

3. **Calculating the Displacement**:
- The distance from A to F can be calculated using the geometry of the hexagon. The distance is equal to `2x` (the distance across the hexagon).

4. **Calculate the Total Distance Traveled**:
- The particle moves from A to B, B to C, C to D, D to E, and E to F. This is a total of 5 sides of the hexagon.
- Therefore, the total distance traveled is `5x`.

5. **Calculate the Total Time Taken**:
- The speed of the particle is constant at `v`.
- The time taken to travel the total distance of `5x` can be calculated using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{5x}{v} \]

6. **Calculate the Average Velocity**:
- The average velocity is defined as the total displacement divided by the total time taken.
- Thus, the average velocity \( V_{avg} \) can be calculated as follows:
\[ V_{avg} = \frac{\text{Displacement}}{\text{Time}} = \frac{2x}{\frac{5x}{v}} = \frac{2x \cdot v}{5x} = \frac{2v}{5} \]

### Final Result:
The magnitude of the average velocity of the particle from A to F is \( \frac{2v}{5} \).

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