A person travels along a straight road for the first half with a velocity `v_(1)` and the second half time with a velocity `v_(2)`. Then the mean velocity `vecv` is given by

Options

A. `barv = (v_(1) + v_(2))/(2)`
B. `(2)/(v) = (1)/(v_(1)) + (1)/(v_(2))`
C. `barv = sqrt(v_(1)v_(2))`
D. `barv = sqrt((v_(2))/(v_(1)))`

Explanation

To find the mean velocity of a person traveling along a straight road for the first half of the time with a velocity \( v_1 \) and for the second half of the time with a velocity \( v_2 \), we can follow these steps:

### Step-by-Step Solution:

1. **Define the Total Time**:
Let's denote the total time of travel as \( T \). According to the problem, the person travels for half of this time \( T \) with velocity \( v_1 \) and the other half with velocity \( v_2 \). Therefore, the time spent at each velocity is:
\[ t_1 = \frac{T}{2} \quad \text{(time at velocity } v_1\text{)} \]
\[ t_2 = \frac{T}{2} \quad \text{(time at velocity } v_2\text{)} \]

2. **Calculate the Distances**:
The distance covered during the first half of the time can be calculated using the formula:
\[ s_1 = v_1 \cdot t_1 = v_1 \cdot \frac{T}{2} \]
The distance covered during the second half of the time is:
\[ s_2 = v_2 \cdot t_2 = v_2 \cdot \frac{T}{2} \]

3. **Total Distance**:
The total distance \( S \) traveled by the person is the sum of the distances covered in both halves:
\[ S = s_1 + s_2 = v_1 \cdot \frac{T}{2} + v_2 \cdot \frac{T}{2} \]
Simplifying this, we get:
\[ S = \frac{T}{2} (v_1 + v_2) \]

4. **Mean Velocity Calculation**:
The mean velocity \( v_m \) is defined as the total distance traveled divided by the total time taken:
\[ v_m = \frac{S}{T} = \frac{\frac{T}{2} (v_1 + v_2)}{T} \]
Simplifying this expression, we find:
\[ v_m = \frac{v_1 + v_2}{2} \]

### Final Answer:
Thus, the mean velocity \( \vec{v} \) is given by:
\[ \vec{v} = \frac{v_1 + v_2}{2} \]