A particle is rotating in a cirlce of radius 1 m with constant speed 4 m/s. In time 1s, match the following (in `SI` unit)
`{:(,"Table-1",,"Table-2"),("(A)","Displacement","(P)","8 sin 2"),("(B)","Distance","(Q)",4),("(C)","Average velocity","(R)","2 sin 2"),("(D)","Average acceleration","(S)","4 sin 2"):}`

Answer

To solve the problem, we will analyze the motion of a particle rotating in a circle of radius 1 m with a constant speed of 4 m/s over a time interval of 1 second. We need to find the displacement, distance, average velocity, and average acceleration, and then match them with the given options.

### Step-by-Step Solution:

1. **Calculate Angular Velocity (ω)**:
The relationship between linear velocity (v), radius (r), and angular velocity (ω) is given by:
\[ v = r \cdot \omega \]
Given \( v = 4 \, \text{m/s} \) and \( r = 1 \, \text{m} \):
\[ \omega = \frac{v}{r} = \frac{4}{1} = 4 \, \text{rad/s} \]

**Hint**: Remember that angular velocity is the rate of change of angular displacement.

2. **Calculate Angular Displacement (θ)**:
The angular displacement can be calculated using:
\[ \theta = \omega \cdot t \]
Given \( t = 1 \, \text{s} \):
\[ \theta = 4 \cdot 1 = 4 \, \text{rad} \]

**Hint**: Angular displacement is the angle through which the particle has rotated in a given time.

3. **Calculate Distance Traveled**:
The distance traveled by the particle along the circular path is equal to the arc length, which can be calculated as:
\[ \text{Distance} = r \cdot \theta \]
Substituting the values:
\[ \text{Distance} = 1 \cdot 4 = 4 \, \text{m} \]

**Hint**: The distance is the length of the path traveled, which is the arc length in circular motion.

4. **Calculate Displacement**:
The displacement in circular motion can be found using the formula:
\[ \text{Displacement} = 2 \cdot r \cdot \sin\left(\frac{\theta}{2}\right) \]
Substituting the values:
\[ \text{Displacement} = 2 \cdot 1 \cdot \sin\left(\frac{4}{2}\right) = 2 \cdot \sin(2) = 2 \sin(2) \]

**Hint**: Displacement is the straight-line distance from the starting point to the endpoint.

5. **Calculate Average Velocity**:
Average velocity is defined as:
\[ \text{Average Velocity} = \frac{\text{Displacement}}{t} \]
Substituting the values:
\[ \text{Average Velocity} = \frac{2 \sin(2)}{1} = 2 \sin(2) \, \text{m/s} \]

**Hint**: Average velocity considers the straight-line displacement over time.

6. **Calculate Average Acceleration**:
Average acceleration can be calculated as:
\[ \text{Average Acceleration} = \frac{\text{Change in Velocity}}{t} \]
In circular motion, the change in velocity can be related to the average velocity:
\[ \text{Average Acceleration} = \frac{2 \cdot 2 \sin(2)}{1} = 4 \sin(2) \, \text{m/s}^2 \]

**Hint**: Average acceleration is the change in velocity over the time interval.

### Matching Results with Options:
- (A) Displacement → (R) \( 2 \sin 2 \)
- (B) Distance → (Q) \( 4 \)
- (C) Average Velocity → (R) \( 2 \sin 2 \)
- (D) Average Acceleration → (S) \( 4 \sin 2 \)

### Final Matching:
- (A) Displacement → (R) \( 2 \sin 2 \)
- (B) Distance → (Q) \( 4 \)
- (C) Average Velocity → (R) \( 2 \sin 2 \)
- (D) Average Acceleration → (S) \( 4 \sin 2 \)