A particle moves in a circle of radius `R = 21/22 m` with constant speed `1 m//s.` Find,
(a) magnitude of average velocity and
(b) magnitude of average acceleration in 2 s.

Explanation

To solve the problem step by step, we will find the magnitude of average velocity and average acceleration of a particle moving in a circle with the given parameters.

### Given:
- Radius \( R = \frac{21}{22} \, \text{m} \)
- Constant speed \( v = 1 \, \text{m/s} \)
- Time \( t = 2 \, \text{s} \)

### (a) Magnitude of Average Velocity

1. **Calculate the Circumference of the Circle**:
\[ \text{Circumference} = 2\pi R = 2\pi \left(\frac{21}{22}\right) = \frac{42\pi}{22} \approx 6.0 \, \text{m} \]

2. **Calculate the Time Period of Motion**:
\[ \text{Time Period} (T) = \frac{\text{Circumference}}{\text{Speed}} = \frac{6.0}{1} = 6 \, \text{s} \]

3. **Determine the Fraction of the Time Period Covered in 2 Seconds**:
\[ \text{Fraction of Time Period} = \frac{t}{T} = \frac{2}{6} = \frac{1}{3} \]

4. **Calculate the Angle Covered in 2 Seconds**:
\[ \text{Angle} = 2\pi \times \frac{1}{3} = \frac{2\pi}{3} \, \text{radians} = 120^\circ \]

5. **Calculate the Displacement**:
The displacement for a circular motion can be calculated using the formula:
\[ \text{Displacement} = 2R \sin\left(\frac{\theta}{2}\right) \]
where \( \theta = 120^\circ \):
\[ \text{Displacement} = 2 \left(\frac{21}{22}\right) \sin\left(60^\circ\right) = 2 \left(\frac{21}{22}\right) \left(\frac{\sqrt{3}}{2}\right) = \frac{21\sqrt{3}}{22} \]

6. **Calculate Average Velocity**:
\[ \text{Average Velocity} = \frac{\text{Displacement}}{t} = \frac{\frac{21\sqrt{3}}{22}}{2} = \frac{21\sqrt{3}}{44} \, \text{m/s} \]

### (b) Magnitude of Average Acceleration

1. **Determine Initial and Final Velocity Vectors**:
- Initial velocity \( \vec{v_i} \) is directed tangentially at the starting point.
- Final velocity \( \vec{v_f} \) is directed tangentially at the point after 120° rotation.

2. **Calculate the Change in Velocity**:
The magnitude of the change in velocity can be calculated using the formula:
\[ |\Delta \vec{v}| = \sqrt{v^2 + v^2 - 2v^2 \cos(\theta)} \]
where \( \theta = 120^\circ \):
\[ |\Delta \vec{v}| = \sqrt{1^2 + 1^2 - 2 \cdot 1 \cdot 1 \cdot \left(-\frac{1}{2}\right)} = \sqrt{2 + 1} = \sqrt{3} \]

3. **Calculate Average Acceleration**:
\[ \text{Average Acceleration} = \frac{|\Delta \vec{v}|}{t} = \frac{\sqrt{3}}{2} \, \text{m/s}^2 \]

### Final Answers:
- (a) Magnitude of Average Velocity: \( \frac{21\sqrt{3}}{44} \, \text{m/s} \)
- (b) Magnitude of Average Acceleration: \( \frac{\sqrt{3}}{2} \, \text{m/s}^2 \)