A car completes its journey in a straight line in three equal parts with speeds `v_1,v_2 and v_3` respectively. The average speed `v` is given by :

Options

A. `(v_1 + v_2 + v_3)/(3)`
B. `3 sqrt(v_1 v_2 v_3)`
C. `(1)/(v)=(1)/(v_1)+(1)/(v_2)+(1)/(v_3)`
D. `(3)/(v)=(1)/(v_1)+(1)/(v_2)+(1)/(v_3)`

Explanation

To find the average speed of a car that completes its journey in three equal parts with speeds \( v_1, v_2, \) and \( v_3 \), we can follow these steps:

### Step-by-Step Solution:

1. **Define the Total Distance**:
Since the journey is divided into three equal parts, let each part have a distance \( L \). Therefore, the total distance \( D \) is:
\[ D = L + L + L = 3L \]

2. **Calculate the Time Taken for Each Part**:
The time taken to cover each part of the journey can be calculated using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
Thus, the times for each part are:
- For the first part:
\[ t_1 = \frac{L}{v_1} \]
- For the second part:
\[ t_2 = \frac{L}{v_2} \]
- For the third part:
\[ t_3 = \frac{L}{v_3} \]

3. **Calculate the Total Time Taken**:
The total time \( T \) taken for the entire journey is the sum of the times for each part:
\[ T = t_1 + t_2 + t_3 = \frac{L}{v_1} + \frac{L}{v_2} + \frac{L}{v_3} \]

4. **Express Total Time in Terms of L**:
We can factor out \( L \) from the total time:
\[ T = L \left( \frac{1}{v_1} + \frac{1}{v_2} + \frac{1}{v_3} \right) \]

5. **Calculate the Average Speed**:
The average speed \( v \) is defined as the total distance divided by the total time:
\[ v = \frac{D}{T} = \frac{3L}{L \left( \frac{1}{v_1} + \frac{1}{v_2} + \frac{1}{v_3} \right)} \]
Simplifying this gives:
\[ v = \frac{3}{\left( \frac{1}{v_1} + \frac{1}{v_2} + \frac{1}{v_3} \right)} \]

### Final Result:
Thus, the average speed \( v \) of the car is:
\[ v = \frac{3}{\left( \frac{1}{v_1} + \frac{1}{v_2} + \frac{1}{v_3} \right)} \]