A man runs along the straight road for half the distance with velocity `v_(1)` and the remaining half distance with velocity `v_(2)`. Then the average velocity is given by

Options

A. `v_(1)v_(2)`
B. `v_(2)^(2)//v_(1)^(2)`
C. `(v_(1) + v_(2))//2`
D. `2v_(1)v_(2)//(v_(1) + v_(2))`

Answer

To find the average velocity of a man running half the distance with velocity \( v_1 \) and the other half with velocity \( v_2 \), we can follow these steps:

### Step-by-Step Solution

1. **Define the Total Distance**:
Let the total distance be \( 2x \). This means the man runs half the distance \( x \) with velocity \( v_1 \) and the other half \( x \) with velocity \( v_2 \).

2. **Calculate Time for Each Half**:
- For the first half (distance \( x \) with velocity \( v_1 \)):
\[ t_1 = \frac{x}{v_1} \]
- For the second half (distance \( x \) with velocity \( v_2 \)):
\[ t_2 = \frac{x}{v_2} \]

3. **Calculate Total Time**:
The total time \( T \) taken to cover the entire distance \( 2x \) is the sum of the times for both halves:
\[ T = t_1 + t_2 = \frac{x}{v_1} + \frac{x}{v_2} \]

4. **Combine the Times**:
To combine the two fractions, we can find a common denominator:
\[ T = x \left( \frac{1}{v_1} + \frac{1}{v_2} \right) = x \left( \frac{v_2 + v_1}{v_1 v_2} \right) \]

5. **Calculate Average Velocity**:
The average velocity \( V_{avg} \) is defined as the total distance divided by the total time:
\[ V_{avg} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{2x}{T} \]
Substituting the expression for \( T \):
\[ V_{avg} = \frac{2x}{x \left( \frac{v_1 + v_2}{v_1 v_2} \right)} \]

6. **Simplify the Expression**:
The \( x \) in the numerator and denominator cancels out:
\[ V_{avg} = \frac{2}{\frac{v_1 + v_2}{v_1 v_2}} = \frac{2 v_1 v_2}{v_1 + v_2} \]

### Final Answer
Thus, the average velocity of the man is:
\[ V_{avg} = \frac{2 v_1 v_2}{v_1 + v_2} \]