If a body covers first half of its journey with uniform speed `v_(1)` and the second half of the journey with uniform speed `v_(2)`. Then the average speed is

Options

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

Explanation

To find the average speed of a body that covers the first half of its journey with uniform speed \( v_1 \) and the second half with uniform speed \( v_2 \), we can follow these steps:

### Step-by-Step Solution:

1. **Define the Total Distance**:
Let the total distance of the journey be \( S \). Therefore, the first half of the journey is \( \frac{S}{2} \) and the second half is also \( \frac{S}{2} \).

2. **Calculate Time for Each Half**:
- For the first half of the journey (distance \( \frac{S}{2} \) at speed \( v_1 \)):
\[ T_1 = \frac{\text{Distance}}{\text{Speed}} = \frac{\frac{S}{2}}{v_1} = \frac{S}{2v_1} \]
- For the second half of the journey (distance \( \frac{S}{2} \) at speed \( v_2 \)):
\[ T_2 = \frac{\text{Distance}}{\text{Speed}} = \frac{\frac{S}{2}}{v_2} = \frac{S}{2v_2} \]

3. **Calculate Total Time**:
The total time \( T \) taken for the journey is the sum of \( T_1 \) and \( T_2 \):
\[ T = T_1 + T_2 = \frac{S}{2v_1} + \frac{S}{2v_2} \]
To simplify this, we can factor out \( \frac{S}{2} \):
\[ T = \frac{S}{2} \left( \frac{1}{v_1} + \frac{1}{v_2} \right) \]

4. **Calculate Average Speed**:
The average speed \( V_{\text{avg}} \) is defined as the total distance divided by the total time:
\[ V_{\text{avg}} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{S}{T} \]
Substituting the expression for \( T \):
\[ V_{\text{avg}} = \frac{S}{\frac{S}{2} \left( \frac{1}{v_1} + \frac{1}{v_2} \right)} \]
Simplifying this gives:
\[ V_{\text{avg}} = \frac{S \cdot 2}{S \left( \frac{1}{v_1} + \frac{1}{v_2} \right)} = \frac{2}{\left( \frac{1}{v_1} + \frac{1}{v_2} \right)} \]

5. **Final Expression**:
The expression can be rewritten using the formula for the harmonic mean:
\[ V_{\text{avg}} = \frac{2v_1v_2}{v_1 + v_2} \]

### Conclusion:
Thus, the average speed of the body for the entire journey is:
\[ V_{\text{avg}} = \frac{2v_1v_2}{v_1 + v_2} \]