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, we can use the formula for average speed when the distances covered are different. The average speed \( V_{avg} \) can be calculated using the formula:

\[ V_{avg} = \frac{\text{Total Distance}}{\text{Total Time}} \]

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

### Step 2: Calculate the distances covered
The car covers:
- \(\frac{2}{5}D\) with speed \( v_1 \)
- \(\frac{3}{5}D\) with speed \( v_2 \)

### Step 3: Calculate the time taken for each segment
The time taken to cover the first segment is given by:
\[ t_1 = \frac{\text{Distance}}{\text{Speed}} = \frac{\frac{2}{5}D}{v_1} = \frac{2D}{5v_1} \]

The time taken to cover the second segment is:
\[ t_2 = \frac{\frac{3}{5}D}{v_2} = \frac{3D}{5v_2} \]

### Step 4: Calculate the total time
The total time \( T \) taken for the journey is:
\[ T = t_1 + t_2 = \frac{2D}{5v_1} + \frac{3D}{5v_2} \]

### Step 5: Substitute into the average speed formula
Now, substituting the total distance and total time into the average speed formula:
\[ V_{avg} = \frac{D}{T} = \frac{D}{\frac{2D}{5v_1} + \frac{3D}{5v_2}} \]

### Step 6: Simplify the expression
We can factor out \( D \) from the denominator:
\[ V_{avg} = \frac{1}{\frac{2}{5v_1} + \frac{3}{5v_2}} = \frac{1}{\frac{2v_2 + 3v_1}{5v_1v_2}} = \frac{5v_1v_2}{2v_2 + 3v_1} \]

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

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