The position vector of a particle `vec(R )` as a funtion of time is given by:
`vec(R )= 4sin(2pit)hat(i)+4cos(2pit)hat(j)`
Where `R` is in meters, `t` is in seconds and `hat(i)` and `hat(j)` denote until vectors along x-and y- directions, respectively Which one of the following statements is wrong for the motion of particle ?

Options

A. Acceleration is along `vec(-R )`
B. Magnitude of accleration vector is `(v^(2))/(R ).`
C. path of the particle is a circle of radius 4 m
D. Magnitude of velocity of particle is 8 m/s

Explanation

To solve the problem, we need to analyze the given position vector of the particle and calculate its velocity and acceleration. Then, we will evaluate the statements provided to determine which one is incorrect.

### Step 1: Write down the position vector
The position vector of the particle is given as:
\[ \vec{R}(t) = 4 \sin(2\pi t) \hat{i} + 4 \cos(2\pi t) \hat{j} \]

### Step 2: Calculate the velocity vector
To find the velocity vector, we differentiate the position vector with respect to time \( t \):
\[ \vec{V}(t) = \frac{d\vec{R}}{dt} = \frac{d}{dt}(4 \sin(2\pi t) \hat{i} + 4 \cos(2\pi t) \hat{j}) \]
Using the chain rule:
\[ \vec{V}(t) = 4 \cdot 2\pi \cos(2\pi t) \hat{i} - 4 \cdot 2\pi \sin(2\pi t) \hat{j} \]
\[ \vec{V}(t) = 8\pi \cos(2\pi t) \hat{i} - 8\pi \sin(2\pi t) \hat{j} \]

### Step 3: Calculate the magnitude of the velocity
The magnitude of the velocity vector is given by:
\[ |\vec{V}| = \sqrt{(8\pi \cos(2\pi t))^2 + (-8\pi \sin(2\pi t))^2} \]
\[ |\vec{V}| = \sqrt{64\pi^2 (\cos^2(2\pi t) + \sin^2(2\pi t))} \]
Using the Pythagorean identity \(\cos^2(x) + \sin^2(x) = 1\):
\[ |\vec{V}| = 8\pi \text{ m/s} \]

### Step 4: Calculate the acceleration vector
Next, we differentiate the velocity vector to find the acceleration vector:
\[ \vec{A}(t) = \frac{d\vec{V}}{dt} = \frac{d}{dt}(8\pi \cos(2\pi t) \hat{i} - 8\pi \sin(2\pi t) \hat{j}) \]
\[ \vec{A}(t) = -8\pi \cdot 2\pi \sin(2\pi t) \hat{i} - 8\pi \cdot 2\pi \cos(2\pi t) \hat{j} \]
\[ \vec{A}(t) = -16\pi^2 \sin(2\pi t) \hat{i} - 16\pi^2 \cos(2\pi t) \hat{j} \]

### Step 5: Calculate the magnitude of the acceleration
The magnitude of the acceleration vector is:
\[ |\vec{A}| = \sqrt{(-16\pi^2 \sin(2\pi t))^2 + (-16\pi^2 \cos(2\pi t))^2} \]
\[ |\vec{A}| = 16\pi^2 \sqrt{\sin^2(2\pi t) + \cos^2(2\pi t)} = 16\pi^2 \text{ m/s}^2 \]

### Step 6: Analyze the statements
Now, we need to check the statements regarding the motion of the particle:
1. The magnitude of the velocity of the particle is \(8\) m/s. (This is incorrect, as we found it to be \(8\pi\) m/s.)
2. The magnitude of the acceleration is \(16\pi^2\) m/s². (This is correct.)
3. The path of the particle is circular. (This is correct, as shown by the equation \(x^2 + y^2 = 16\).)

### Conclusion
The incorrect statement is that the magnitude of the velocity of the particle is \(8\) m/s. The correct value is \(8\pi\) m/s.