A man starts walking on a circular track of radius `R`. First half of the distance he walks with speed `V_(1)` , half of the remaining distance with speed `V_(2)`, then half of the remaining time with `V_(1)` and rest with `V_(2)` and completes the circle. Average speed of the man during entire motion in which he completes the circle is.

Options

A. `(2V_(1)V_(2)(V_(1)+V_(2)))/(V_(2)^(2)+2V_(1)^(2)+2V_(1)V_(2))`
B. `(4V_(1)V_(2)(V_(1)+V_(2)))/(V_(1)^(2)+2V_(2)^(2)+5V_(1)V_(2))`
C. `(V_(1)V_(2)(V_(1)+2V_(2)))/(V_(1)^(2)+2V_(2)^(2)+5V_(1)V_(2))`
D. `((V_(1)+2V_(2))^(2))/(V_(1)+V_(2)+2V_(1)^(2)V_(2)^(2))`

Explanation

To solve the problem, we need to find the average speed of a man completing a circular track of radius \( R \) while walking at different speeds during different segments of the journey.

### Step-by-Step Solution:

1. **Calculate the Total Distance:**
The total distance around the circular track is given by the circumference:
\[ D = 2\pi R \]

2. **First Half of the Distance:**
The man walks the first half of the distance with speed \( V_1 \):
\[ \text{Distance}_1 = \frac{D}{2} = \frac{2\pi R}{2} = \pi R \]
The time taken for this distance is:
\[ t_1 = \frac{\text{Distance}_1}{V_1} = \frac{\pi R}{V_1} \]

3. **Half of the Remaining Distance:**
The remaining distance after the first half is also \( \pi R \). The man walks half of this remaining distance with speed \( V_2 \):
\[ \text{Distance}_2 = \frac{\pi R}{2} \]
The time taken for this distance is:
\[ t_2 = \frac{\text{Distance}_2}{V_2} = \frac{\pi R/2}{V_2} = \frac{\pi R}{2 V_2} \]

4. **Remaining Distance and Time:**
After walking \( \pi R/2 \), the remaining distance is \( \pi R/2 \). He walks half of the remaining time with speed \( V_1 \) and the rest with speed \( V_2 \). Let the total time for this segment be \( t_0 \). Half of the remaining time is \( t_0/2 \).

The distance covered in the first half of this time with speed \( V_1 \) is:
\[ \text{Distance}_3 = V_1 \cdot \frac{t_0}{2} \]
The distance covered in the second half of this time with speed \( V_2 \) is:
\[ \text{Distance}_4 = V_2 \cdot \frac{t_0}{2} \]
The sum of these distances must equal the remaining distance:
\[ V_1 \cdot \frac{t_0}{2} + V_2 \cdot \frac{t_0}{2} = \frac{\pi R}{2} \]
Simplifying gives:
\[ \frac{t_0}{2} (V_1 + V_2) = \frac{\pi R}{2} \]
Thus, we find:
\[ t_0 = \frac{\pi R}{V_1 + V_2} \]

5. **Total Time Calculation:**
The total time \( T \) taken for the entire journey is:
\[ T = t_1 + t_2 + t_0 = \frac{\pi R}{V_1} + \frac{\pi R}{2 V_2} + \frac{\pi R}{V_1 + V_2} \]

6. **Average Speed Calculation:**
The average speed \( V_{avg} \) is given by the total distance divided by the total time:
\[ V_{avg} = \frac{D}{T} = \frac{2\pi R}{\frac{\pi R}{V_1} + \frac{\pi R}{2 V_2} + \frac{\pi R}{V_1 + V_2}} \]
Simplifying this expression:
\[ V_{avg} = \frac{2}{\frac{1}{V_1} + \frac{1}{2 V_2} + \frac{1}{V_1 + V_2}} \]

7. **Final Expression:**
After finding a common denominator and simplifying, we arrive at the final expression for average speed:
\[ V_{avg} = \frac{4 V_1 V_2 (V_1 + V_2)}{V_1^2 + 2 V_2^2 + 5 V_1 V_2} \]