If a car covers 2/5th of the total distance with `v_1` speed and 3/5th 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)`

Answer

To solve the problem of finding the average speed of a car that covers different fractions of a total distance at different speeds, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Total Distance**:
Let's assume the total distance covered by the car is \( D \). For simplicity, we can take \( D = 5x \) where \( x \) is a unit distance.

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

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

4. **Calculate the Total Time**:
The total time taken for the journey is the sum of the times for both segments:
\[ \text{Total Time} = \text{Time}_1 + \text{Time}_2 = \frac{2x}{v_1} + \frac{3x}{v_2} \]

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

6. **Further Simplification**:
To combine the fractions in the denominator:
\[ V_{avg} = \frac{5}{\frac{2v_2 + 3v_1}{v_1 v_2}} = \frac{5v_1 v_2}{2v_2 + 3v_1} \]

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