A particle covers half of its total distance with speed `v_1` and the rest half distance with speed `v_2`. Its average speed during the complete journey is.

Options

A. `(v_(1)+v_(2))/(2)`
B. `(v_(1)v_(2))/(v_(1)+v_(2))`
C. `(2v_(1)v_(2))/(v_(1)+v_(2))`
D. `(v_1^2v_2^2)/(v_1^2+v_2^2)`

Explanation

To find the average speed of a particle that covers half of its total distance with speed \( v_1 \) and the other half with speed \( v_2 \), we can follow these steps:

### Step-by-Step Solution:

1. **Define Total Distance**:
Let the total distance covered by the particle be \( D \). Since the particle covers half of this distance with speed \( v_1 \) and the other half with speed \( v_2 \), we can express the distances as:
- Distance covered with speed \( v_1 \): \( \frac{D}{2} \)
- Distance covered with speed \( v_2 \): \( \frac{D}{2} \)

2. **Calculate Time for Each Segment**:
- Time taken to cover the first half of the distance:
\[ t_1 = \frac{\text{Distance}}{\text{Speed}} = \frac{\frac{D}{2}}{v_1} = \frac{D}{2v_1} \]
- Time taken to cover the second half of the distance:
\[ t_2 = \frac{\frac{D}{2}}{v_2} = \frac{D}{2v_2} \]

3. **Calculate Total Time**:
The total time \( T \) taken for the entire journey is the sum of \( t_1 \) and \( t_2 \):
\[ T = t_1 + t_2 = \frac{D}{2v_1} + \frac{D}{2v_2} \]
To combine these fractions, we can find a common denominator:
\[ T = \frac{D}{2} \left( \frac{1}{v_1} + \frac{1}{v_2} \right) = \frac{D}{2} \cdot \frac{v_1 + v_2}{v_1 v_2} \]

4. **Calculate Average Speed**:
The average speed \( V_{avg} \) is defined as the total distance divided by the total time:
\[ V_{avg} = \frac{D}{T} \]
Substituting \( T \):
\[ V_{avg} = \frac{D}{\frac{D}{2} \cdot \frac{v_1 + v_2}{v_1 v_2}} = \frac{D \cdot 2 \cdot v_1 v_2}{D \cdot (v_1 + v_2)} \]
The \( D \) cancels out:
\[ V_{avg} = \frac{2 v_1 v_2}{v_1 + v_2} \]

### Final Answer:
The average speed during the complete journey is:
\[ V_{avg} = \frac{2 v_1 v_2}{v_1 + v_2} \]