A person travels along a straight road for the first half time with a velocity `v_1 `and the next half time with a velocity `v_2` . The mean velocity V of the man is

Options

A. `2/v = (1)/(v_1) + (1)/(v_2)`
B. `V = (v_1 + v_2)/(2)`
C. `V = sqrt(v_1v_2)`
D. `V = sqrt(v_1/v_2)`

Explanation

To find the mean velocity \( V \) of a person who travels 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 assume the total time of travel is \( 2T \). This means the first half of the time is \( T \) and the second half is also \( T \).

2. **Calculate Displacement for Each Half**:
- For the first half of the time (duration \( T \)), the displacement \( D_1 \) can be calculated using the formula:
\[ D_1 = v_1 \times T \]
- For the second half of the time (duration \( T \)), the displacement \( D_2 \) is:
\[ D_2 = v_2 \times T \]

3. **Calculate Total Displacement**:
The total displacement \( D \) is the sum of the displacements from both halves:
\[ D = D_1 + D_2 = v_1 \times T + v_2 \times T = (v_1 + v_2) \times T \]

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

5. **Calculate Mean Velocity**:
The mean velocity \( V \) is defined as the total displacement divided by the total time:
\[ V = \frac{D}{\text{Total Time}} = \frac{(v_1 + v_2) \times T}{2T} \]
Simplifying this gives:
\[ V = \frac{v_1 + v_2}{2} \]

### Final Answer:
The mean velocity \( V \) of the man is:
\[ V = \frac{v_1 + v_2}{2} \]