If a car covers `(2)/(5)^(th)` of the total distance with `v_1` speed and `(3)/(5)^(th)` distance with `v_2`. Then average speed is

Options

A. ` (1)/( 2) srqt(v_1v_2)`
B. ` (v_1+v_2)/( 2) `
C. ` (2v_1v_2)/( v_1+v_2) `
D. ` (5v_1v_2)/( 3v_1+2v_2) `

Explanation

To find the average speed of the car that covers different portions of a total distance with different speeds, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Total Distance**:
Let the total distance be \( x \).

2. **Calculate the Distances Covered**:
- The car covers \( \frac{2}{5} \) of the total distance with speed \( v_1 \):
\[ \text{Distance}_1 = \frac{2}{5} x \]
- The car covers \( \frac{3}{5} \) of the total distance with speed \( v_2 \):
\[ \text{Distance}_2 = \frac{3}{5} x \]

3. **Calculate the Time Taken for Each Segment**:
- The time taken to cover the first segment is given by:
\[ T_1 = \frac{\text{Distance}_1}{v_1} = \frac{\frac{2}{5} x}{v_1} = \frac{2x}{5v_1} \]
- The time taken to cover the second segment is given by:
\[ T_2 = \frac{\text{Distance}_2}{v_2} = \frac{\frac{3}{5} x}{v_2} = \frac{3x}{5v_2} \]

4. **Calculate the Total Time Taken**:
The total time taken \( T \) is the sum of \( T_1 \) and \( T_2 \):
\[ T = T_1 + T_2 = \frac{2x}{5v_1} + \frac{3x}{5v_2} \]

5. **Simplify the Total Time**:
Factor out \( x \):
\[ T = x \left( \frac{2}{5v_1} + \frac{3}{5v_2} \right) = \frac{x}{5} \left( \frac{2}{v_1} + \frac{3}{v_2} \right) \]

6. **Calculate the Average Speed**:
The average speed \( V_{avg} \) is defined as the total distance divided by the total time:
\[ V_{avg} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{x}{T} \]
Substituting for \( T \):
\[ V_{avg} = \frac{x}{\frac{x}{5} \left( \frac{2}{v_1} + \frac{3}{v_2} \right)} = \frac{5}{\frac{2}{v_1} + \frac{3}{v_2}} \]

7. **Finding a Common Denominator**:
To simplify further, we can find a common denominator:
\[ V_{avg} = \frac{5 v_1 v_2}{3 v_1 + 2 v_2} \]

8. **Final Result**:
Thus, the average speed of the car is:
\[ V_{avg} = \frac{5 v_1 v_2}{3 v_1 + 2 v_2} \]

### Conclusion:
The average speed of the car is given by the formula:
\[ V_{avg} = \frac{5 v_1 v_2}{3 v_1 + 2 v_2} \]
This matches with option D.