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)/(v_1 + v_2)`
B. `(2 v_1 v_2)/(v_1 + v_2)`
C. `(2 v_1^2 v_2^2)/(v_1^2 + v_2^2)`
D. `(v_1 + 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 1: Define the total distance
Let the total distance covered by the particle be \( 2D \). This means that the first half of the distance is \( D \) and the second half is also \( D \).

### Step 2: Calculate the time taken for the first half of the distance
The time taken to cover the first half of the distance \( D \) with speed \( v_1 \) can be calculated using the formula:
\[ T_1 = \frac{D}{v_1} \]

### Step 3: Calculate the time taken for the second half of the distance
The time taken to cover the second half of the distance \( D \) with speed \( v_2 \) is given by:
\[ T_2 = \frac{D}{v_2} \]

### Step 4: Calculate the total time for the journey
The total time \( T \) taken for the entire journey is the sum of the times for both halves:
\[ T = T_1 + T_2 = \frac{D}{v_1} + \frac{D}{v_2} \]

### Step 5: Simplify the total time expression
We can factor out \( D \) from the total time expression:
\[ T = D \left( \frac{1}{v_1} + \frac{1}{v_2} \right) \]

### Step 6: Calculate the average speed
The average speed \( V_{avg} \) is defined as the total distance divided by the total time:
\[ V_{avg} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{2D}{T} \]
Substituting the expression for \( T \):
\[ V_{avg} = \frac{2D}{D \left( \frac{1}{v_1} + \frac{1}{v_2} \right)} = \frac{2}{\frac{1}{v_1} + \frac{1}{v_2}} \]

### Step 7: Final expression for average speed
Thus, the average speed during the complete journey is given by:
\[ V_{avg} = \frac{2v_1 v_2}{v_1 + v_2} \]

### Summary of the Solution Steps:
1. Define total distance as \( 2D \).
2. Calculate time for first half: \( T_1 = \frac{D}{v_1} \).
3. Calculate time for second half: \( T_2 = \frac{D}{v_2} \).
4. Find total time: \( T = T_1 + T_2 \).
5. Simplify total time expression.
6. Calculate average speed using total distance and total time.
7. Present the final expression for average speed.