A person travels along a straight road for half the distance with velocity `V_1` and the remaining half distance with velocity `V_2` the average velocity is given by

Options

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

Answer

To find the average velocity of a person who travels half the distance with velocity \( V_1 \) and the other half with velocity \( V_2 \), we can follow these steps:

### Step 1: Define Total Distance
Let's assume the total distance traveled by the person is \( 2x \). This means the first half of the distance is \( x \) and the second half is also \( x \).

### Step 2: Calculate Total Displacement
The total displacement for the journey is equal to the total distance traveled, which is:
\[ \text{Total Displacement} = 2x \]

### Step 3: Calculate Total Time Taken
The total time taken to travel the distance can be calculated by adding the time taken for each half of the journey.

1. For the first half (distance \( x \) at velocity \( V_1 \)):
\[ \text{Time}_1 = \frac{x}{V_1} \]

2. For the second half (distance \( x \) at velocity \( V_2 \)):
\[ \text{Time}_2 = \frac{x}{V_2} \]

Now, the total time taken \( T \) is:
\[ \text{Total Time} = \text{Time}_1 + \text{Time}_2 = \frac{x}{V_1} + \frac{x}{V_2} \]

### Step 4: Simplify Total Time
We can factor out \( x \) from the total time:
\[ \text{Total Time} = x \left( \frac{1}{V_1} + \frac{1}{V_2} \right) \]

### Step 5: Calculate Average Velocity
The average velocity \( V_{\text{avg}} \) is given by the formula:
\[ V_{\text{avg}} = \frac{\text{Total Displacement}}{\text{Total Time}} \]
Substituting the values we found:
\[ V_{\text{avg}} = \frac{2x}{x \left( \frac{1}{V_1} + \frac{1}{V_2} \right)} \]

### Step 6: Cancel \( x \) and Simplify
Cancelling \( x \) from the numerator and denominator:
\[ V_{\text{avg}} = \frac{2}{\frac{1}{V_1} + \frac{1}{V_2}} \]

### Step 7: Further Simplification
To simplify this expression, we can find a common denominator:
\[ V_{\text{avg}} = \frac{2V_1V_2}{V_1 + V_2} \]

### Final Answer
Thus, the average velocity of the person is:
\[ V_{\text{avg}} = \frac{2V_1V_2}{V_1 + V_2} \]