A particle travels first half of the total time with speed `v_1` and second half time with speed `v_2.` Find the average speed during the complete journey.

Options

A. `(v_1+v_2)`
B. `(v_1-v_2)/2`
C. `(v_1+v_2)/2`
D. `(v_1*v_2)/2`

Explanation

To find the average speed of a particle that travels the first half of the total time with speed \( v_1 \) and the second half with speed \( v_2 \), we can follow these steps:

### Step 1: Define Total Time
Let the total time of the journey be \( 2T \). This means the first half of the journey takes time \( T \) and the second half also takes time \( T \).

### Step 2: Calculate Distances
Using the formula for distance, which is given by:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]
- For the first half of the journey (time \( T \) at speed \( v_1 \)):
\[ d_1 = v_1 \times T \]
- For the second half of the journey (time \( T \) at speed \( v_2 \)):
\[ d_2 = v_2 \times T \]

### Step 3: Total Distance
The total distance \( D \) covered during the entire journey is the sum of \( d_1 \) and \( d_2 \):
\[ D = d_1 + d_2 = v_1 \times T + v_2 \times T = (v_1 + v_2) \times T \]

### Step 4: Total Time
The total time for the journey is:
\[ \text{Total Time} = 2T \]

### Step 5: Calculate Average Speed
The average speed \( v_{\text{avg}} \) is defined as the total distance divided by the total time:
\[ v_{\text{avg}} = \frac{D}{\text{Total Time}} = \frac{(v_1 + v_2) \times T}{2T} \]
Simplifying this expression, we find:
\[ v_{\text{avg}} = \frac{v_1 + v_2}{2} \]

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