A car travels a distance of 'x' with speed `v_1` and then same distance 'x' with speed `v_2` in the same direction. The average speed of the car is :

Options

A. `(v_1v_2)/(2(v_1+v_2)`
B. `(2x)/(v_1+v_2)`
C. `(2v_1v_2)/(v_1+v_2)`
D. `(v_1+v_2)/2`

Answer

To find the average speed of a car that travels a distance of 'x' with speed `v1` and then the same distance 'x' with speed `v2`, we can follow these steps:

### Step 1: Understand the Concept of Average Speed
The average speed is defined as the total distance traveled divided by the total time taken for the journey.

### Step 2: Calculate Total Distance
The car travels a distance 'x' twice, so the total distance (D) is:
\[ D = x + x = 2x \]

### Step 3: Calculate Time Taken for Each Segment
Using the formula for time, \( t = \frac{s}{v} \), we can calculate the time taken for each segment of the journey:
- Time taken to travel the first distance \( x \) at speed \( v1 \):
\[ t_1 = \frac{x}{v_1} \]
- Time taken to travel the second distance \( x \) at speed \( v2 \):
\[ t_2 = \frac{x}{v_2} \]

### Step 4: Calculate Total Time
Now, we can find the total time (T) taken for the entire journey:
\[ T = t_1 + t_2 = \frac{x}{v_1} + \frac{x}{v_2} \]

### Step 5: Simplify Total Time
We can factor out \( x \) from the total time:
\[ T = x \left( \frac{1}{v_1} + \frac{1}{v_2} \right) \]

### Step 6: Calculate Average Speed
Now we can use the formula for average speed:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \]
Substituting the values we calculated:
\[ \text{Average Speed} = \frac{2x}{x \left( \frac{1}{v_1} + \frac{1}{v_2} \right)} \]

### Step 7: Cancel Out 'x'
The 'x' in the numerator and denominator cancels out:
\[ \text{Average Speed} = \frac{2}{\frac{1}{v_1} + \frac{1}{v_2}} \]

### Step 8: Simplify Further
To simplify this expression, we can find a common denominator:
\[ \text{Average Speed} = \frac{2}{\frac{v_1 + v_2}{v_1 v_2}} = \frac{2 \cdot v_1 v_2}{v_1 + v_2} \]

### Final Answer
Thus, the average speed of the car is:
\[ \text{Average Speed} = \frac{2 v_1 v_2}{v_1 + v_2} \]