A particle travels two and a half revolutions of the circle of radius R in time t. The ratio of the average speed of the particle to the magnitude of the average velocity in this time interval is

Options

A. `(pi)/(2)`
B. `(5pi)/(sqrt2)`
C. `(5pi)/(2)`
D. `(pi)/(5sqrt2)`

Answer

To solve the problem, we need to find the ratio of the average speed of a particle to the magnitude of its average velocity after it travels two and a half revolutions around a circle of radius \( R \) in time \( t \).

### Step-by-Step Solution:

1. **Calculate the Total Distance Traveled:**
- The particle completes 2.5 revolutions.
- The circumference of the circle is given by \( C = 2\pi R \).
- Therefore, the total distance traveled by the particle is:
\[ \text{Total Distance} = 2.5 \times C = 2.5 \times 2\pi R = 5\pi R. \]

2. **Calculate the Average Speed:**
- Average speed is defined as the total distance traveled divided by the total time taken.
- Thus, the average speed \( v_{avg} \) is:
\[ v_{avg} = \frac{\text{Total Distance}}{t} = \frac{5\pi R}{t}. \]

3. **Calculate the Net Displacement:**
- The net displacement is the straight-line distance from the initial position to the final position.
- After 2.5 revolutions, the particle ends up on the opposite side of the circle, which is a distance of \( 2R \) (the diameter of the circle).
- Therefore, the net displacement \( d_{net} \) is:
\[ d_{net} = 2R. \]

4. **Calculate the Average Velocity:**
- Average velocity is defined as the net displacement divided by the total time taken.
- Thus, the average velocity \( v_{avg, vel} \) is:
\[ v_{avg, vel} = \frac{d_{net}}{t} = \frac{2R}{t}. \]

5. **Calculate the Ratio of Average Speed to Average Velocity:**
- Now, we need to find the ratio of average speed to average velocity:
\[ \text{Ratio} = \frac{v_{avg}}{v_{avg, vel}} = \frac{\frac{5\pi R}{t}}{\frac{2R}{t}}. \]
- Simplifying this expression:
\[ \text{Ratio} = \frac{5\pi R}{t} \times \frac{t}{2R} = \frac{5\pi}{2}. \]

### Final Answer:
The ratio of the average speed of the particle to the magnitude of the average velocity in this time interval is:
\[ \frac{5\pi}{2}. \]