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 various points (F, D, C, and B), we will follow these steps:

### Step 1: Understanding Average Velocity
Average velocity is defined as the total displacement divided by the total time taken. Mathematically, it can be expressed as:
\[ \text{Average Velocity} = \frac{\text{Displacement}}{\text{Time}} \]

### Step 2: Calculate Average Velocity from A to F
1. **Displacement from A to F**: The straight-line distance (displacement) from point A to point F in a regular hexagon is equal to the length of two sides of the hexagon, which is \(AF = 2X\).
2. **Distance traveled from A to F**: The particle travels along the sides of the hexagon from A to F, covering 5 sides. Therefore, the distance traveled is \(5X\).
3. **Time taken**: The time taken to travel this distance at constant speed \(v\) is given by:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{5X}{v} \]
4. **Average velocity from A to F**:
\[ V_{AF} = \frac{AF}{\text{Time}} = \frac{2X}{\frac{5X}{v}} = \frac{2X \cdot v}{5X} = \frac{2v}{5} \]

### Step 3: Calculate Average Velocity from A to D
1. **Displacement from A to D**: The straight-line distance from A to D is equal to \(AD = 2X\) (as it covers two sides).
2. **Distance traveled from A to D**: The particle travels 3 sides, so the distance is \(3X\).
3. **Time taken**:
\[ \text{Time} = \frac{3X}{v} \]
4. **Average velocity from A to D**:
\[ V_{AD} = \frac{AD}{\text{Time}} = \frac{2X}{\frac{3X}{v}} = \frac{2X \cdot v}{3X} = \frac{2v}{3} \]

### Step 4: Calculate Average Velocity from A to C
1. **Displacement from A to C**: The straight-line distance from A to C can be calculated using the cosine rule in the triangle formed by points A, B, and C. The angle between sides AB and AC is \(120^\circ\), so:
\[ AC = \sqrt{X^2 + X^2 - 2X^2 \cos(120^\circ)} = \sqrt{2X^2(1 + \frac{1}{2})} = \sqrt{3X^2} = X\sqrt{3} \]
2. **Distance traveled from A to C**: The particle travels 2 sides, so the distance is \(2X\).
3. **Time taken**:
\[ \text{Time} = \frac{2X}{v} \]
4. **Average velocity from A to C**:
\[ V_{AC} = \frac{AC}{\text{Time}} = \frac{X\sqrt{3}}{\frac{2X}{v}} = \frac{X\sqrt{3} \cdot v}{2X} = \frac{\sqrt{3}v}{2} \]

### Step 5: Calculate Average Velocity from A to B
1. **Displacement from A to B**: The straight-line distance from A to B is simply \(AB = X\).
2. **Distance traveled from A to B**: The particle travels 1 side, so the distance is \(X\).
3. **Time taken**:
\[ \text{Time} = \frac{X}{v} \]
4. **Average velocity from A to B**:
\[ V_{AB} = \frac{AB}{\text{Time}} = \frac{X}{\frac{X}{v}} = v \]

### Summary of Results
- Average velocity from A to F: \(\frac{2v}{5}\)
- Average velocity from A to D: \(\frac{2v}{3}\)
- Average velocity from A to C: \(\frac{\sqrt{3}v}{2}\)
- Average velocity from A to B: \(v\)