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. ` ( 2 v _(1) v _(2))/( v _(1) + v _(2))`
C. `(v _(1) v _(2))/(v _(1) + v _(2))`
D. `v _(1) v_(2)`

Answer

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's assume the total distance covered by the body is \( 2x \). This means the first half of the journey is \( x \) and the second half is also \( x \).

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

3. **Total Time Taken**:
The total time \( T \) taken to cover the entire distance 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**:
Factor out \( x \) from the total time:
\[ T = x \left( \frac{1}{v_1} + \frac{1}{v_2} \right) \]

5. **Average Speed Formula**:
The average speed \( V \) is defined as total distance divided by total time:
\[ V = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{2x}{T} \]
Substituting \( T \) from the previous step:
\[ V = \frac{2x}{x \left( \frac{1}{v_1} + \frac{1}{v_2} \right)} \]

6. **Simplify the Expression**:
The \( x \) terms cancel out:
\[ V = \frac{2}{\frac{1}{v_1} + \frac{1}{v_2}} \]

7. **Finding a Common Denominator**:
Rewrite the denominator:
\[ V = \frac{2}{\frac{v_1 + v_2}{v_1 v_2}} = \frac{2 v_1 v_2}{v_1 + v_2} \]

8. **Final Result**:
Thus, the average speed \( V \) is given by:
\[ V = \frac{2 v_1 v_2}{v_1 + v_2} \]

### Conclusion:
The average speed of the body is \( \frac{2 v_1 v_2}{v_1 + v_2} \).

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