A person travels along a straight road for the 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)`

Answer

To find the mean velocity \( V \) of a person who travels along a straight road for half the time with velocity \( V_1 \) and the next half time with velocity \( V_2 \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
The person travels for half the time at velocity \( V_1 \) and for the other half at velocity \( V_2 \). We need to find the mean velocity over the entire journey.

2. **Define Total Displacement**:
Since the person travels for half the time at each velocity, we can denote the total time of travel as \( T \). Thus, the time spent at each velocity is \( \frac{T}{2} \).

3. **Calculate Displacement for Each Segment**:
- For the first half of the time:
\[ \text{Displacement}_1 = V_1 \times \frac{T}{2} \]
- For the second half of the time:
\[ \text{Displacement}_2 = V_2 \times \frac{T}{2} \]

4. **Total Displacement**:
The total displacement \( D \) is the sum of the displacements from both segments:
\[ D = \text{Displacement}_1 + \text{Displacement}_2 = V_1 \times \frac{T}{2} + V_2 \times \frac{T}{2} \]
\[ D = \frac{T}{2} (V_1 + V_2) \]

5. **Total Time**:
The total time \( T \) is simply \( T \).

6. **Mean Velocity Formula**:
The mean velocity \( V \) is given by the formula:
\[ V = \frac{\text{Total Displacement}}{\text{Total Time}} = \frac{D}{T} \]
Substituting the values we found:
\[ V = \frac{\frac{T}{2} (V_1 + V_2)}{T} \]

7. **Simplifying the Expression**:
The \( T \) in the numerator and denominator cancels out:
\[ V = \frac{1}{2} (V_1 + V_2) \]

### Final Result:
The mean velocity \( V \) of the man is:
\[ V = \frac{V_1 + V_2}{2} \]