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 distance traveled in one complete revolution of a circle is given by the circumference, which is \( 2\pi R \).
- For two and a half revolutions, the total distance \( D \) is:
\[ D = 2.5 \times (2\pi R) = 5\pi R \]

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

3. **Calculate the Displacement:**
- The displacement is the shortest distance from the initial position to the final position. After two and a half revolutions, the particle ends up at a point directly opposite its starting point on the circle.
- The displacement \( S \) is equal to the diameter of the circle, which is:
\[ S = 2R \]

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

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

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