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)sqrt(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 segments of a total distance with different speeds, we can follow these steps:

### Step 1: Define the total distance
Let the total distance be \( d \).

### Step 2: Calculate the distance covered at each speed
- The car covers \( \frac{2}{5} \) of the distance with speed \( v_1 \).
- The car covers \( \frac{3}{5} \) of the distance with speed \( v_2 \).

### Step 3: Calculate the time taken for each segment
Using the formula \( \text{time} = \frac{\text{distance}}{\text{speed}} \):
- For the first segment:
\[ t_1 = \frac{\frac{2}{5}d}{v_1} = \frac{2d}{5v_1} \]
- For the second segment:
\[ t_2 = \frac{\frac{3}{5}d}{v_2} = \frac{3d}{5v_2} \]

### Step 4: Calculate the total time taken
The total time \( T \) taken to cover the entire distance is:
\[ T = t_1 + t_2 = \frac{2d}{5v_1} + \frac{3d}{5v_2} \]

### Step 5: Find the average speed
The average speed \( V_{avg} \) is given by the formula:
\[ V_{avg} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{d}{T} \]
Substituting the expression for \( T \):
\[ V_{avg} = \frac{d}{\frac{2d}{5v_1} + \frac{3d}{5v_2}} \]

### Step 6: Simplify the expression
Factor out \( d \) from the denominator:
\[ V_{avg} = \frac{d}{d \left( \frac{2}{5v_1} + \frac{3}{5v_2} \right)} = \frac{1}{\frac{2}{5v_1} + \frac{3}{5v_2}} \]
This can be further simplified:
\[ V_{avg} = \frac{1}{\frac{2v_2 + 3v_1}{5v_1v_2}} = \frac{5v_1v_2}{2v_2 + 3v_1} \]

### Final Result
Thus, the average speed of the car is:
\[ V_{avg} = \frac{5v_1v_2}{3v_1 + 2v_2} \]