A particle starts moving from rest at `t=0` with a tangential acceleration of constant magnitude of `pi m//s^(2)` along a circle of radius 6 m. The value of average acceleration, average velocity and average speed during the first `2 sqrt(3)` s of motion, are respectively :

Options

A. `3sqrt(2)m//s^(2),pi m//s,pisqrt(3)m//s`
B. `pi m//s^(2),2 sqrt(3) m//s, pi sqrt(3) m//s`
C. `pi sqrt(3) m//s^(2),2 sqrt(3) m//s, pi m//s`
D. None of the above

Answer

To solve the problem step by step, we need to find the average acceleration, average velocity, and average speed of a particle moving in a circular path with given parameters.

### Given:
- Tangential acceleration, \( a_t = \pi \, \text{m/s}^2 \)
- Radius of the circle, \( r = 6 \, \text{m} \)
- Time, \( t = 2\sqrt{3} \, \text{s} \)
- Initial velocity, \( u = 0 \, \text{m/s} \) (since the particle starts from rest)

### Step 1: Calculate Average Acceleration
The average acceleration \( a_{avg} \) is defined as the change in velocity over time:

\[ a_{avg} = \frac{\Delta v}{\Delta t} \]

First, we need to find the final velocity \( v \) after time \( t \):

Using the equation of motion:

\[ v = u + a_t \cdot t \]

Substituting the values:

\[ v = 0 + \pi \cdot (2\sqrt{3}) = 2\pi\sqrt{3} \, \text{m/s} \]

Now, substituting back to find average acceleration:

\[ a_{avg} = \frac{v - u}{t} = \frac{2\pi\sqrt{3} - 0}{2\sqrt{3}} = \pi \, \text{m/s}^2 \]

### Step 2: Calculate Average Velocity
The average velocity \( v_{avg} \) is defined as the total displacement over time:

\[ v_{avg} = \frac{\text{Total Displacement}}{\text{Time}} \]

To find the total displacement, we first need to calculate the angular displacement \( \theta \):

Using the formula for angular displacement:

\[ \theta = \omega_0 t + \frac{1}{2} \alpha t^2 \]

Since the initial angular velocity \( \omega_0 = 0 \), we have:

\[ \theta = \frac{1}{2} \cdot \frac{a_t}{r} \cdot t^2 \]

Substituting the values:

\[ \theta = \frac{1}{2} \cdot \frac{\pi}{6} \cdot (2\sqrt{3})^2 = \frac{1}{2} \cdot \frac{\pi}{6} \cdot 12 = \pi \, \text{radians} \]

The total displacement in circular motion is the straight line distance between the starting point and the endpoint, which is the diameter of the circle for half a revolution:

\[ \text{Total Displacement} = 2r = 2 \cdot 6 = 12 \, \text{m} \]

Now, substituting in the average velocity formula:

\[ v_{avg} = \frac{12}{2\sqrt{3}} = \frac{12}{2\sqrt{3}} = 2\sqrt{3} \, \text{m/s} \]

### Step 3: Calculate Average Speed
The average speed \( v_{avg, speed} \) is defined as the total distance traveled over time:

The total distance traveled is the arc length for half the circle:

\[ \text{Total Distance} = \frac{1}{2} \cdot 2\pi r = \pi r = \pi \cdot 6 = 6\pi \, \text{m} \]

Now, substituting in the average speed formula:

\[ v_{avg, speed} = \frac{6\pi}{2\sqrt{3}} = \frac{3\pi}{\sqrt{3}} = \pi\sqrt{3} \, \text{m/s} \]

### Final Results:
- Average Acceleration: \( \pi \, \text{m/s}^2 \)
- Average Velocity: \( 2\sqrt{3} \, \text{m/s} \)
- Average Speed: \( \pi\sqrt{3} \, \text{m/s} \)