A particle moves in a circle of radius 1 m with speed of 1 m/s. After completing half cycle. Find out :-
(i) Displacement
(ii) Distance travelled
(iii) Average velocity
(iv) Average acceleration

Answer

To solve the problem step by step, we will analyze the motion of a particle moving in a circle of radius 1 meter with a speed of 1 m/s after completing half a cycle.

### Step 1: Calculate the Distance Travelled
The particle moves in a circular path. The distance travelled by the particle after completing half a cycle (which is half the circumference of the circle) can be calculated using the formula for the circumference of a circle:

\[ \text{Circumference} = 2\pi r \]

For a radius \( r = 1 \) m:

\[ \text{Circumference} = 2\pi \times 1 = 2\pi \, \text{m} \]

Since the particle completes half a cycle, the distance travelled is:

\[ \text{Distance travelled} = \frac{1}{2} \times 2\pi = \pi \, \text{m} \]

### Step 2: Calculate the Displacement
Displacement is defined as the shortest distance between the initial and final positions of the particle. After completing half a cycle, the particle moves from one point on the circle to the point directly opposite it.

Since the radius of the circle is 1 m, the displacement can be calculated as the diameter of the circle:

\[ \text{Displacement} = 2r = 2 \times 1 = 2 \, \text{m} \]

### Step 3: Calculate the Average Velocity
Average velocity is defined as the total displacement divided by the total time taken.

We already calculated:
- Displacement = 2 m

Now, we need to calculate the time taken to complete half a cycle. The time taken \( t \) can be calculated using the formula:

\[ t = \frac{\text{Distance travelled}}{\text{Speed}} \]

Substituting the values:

\[ t = \frac{\pi}{1} = \pi \, \text{s} \]

Now, we can calculate the average velocity:

\[ \text{Average Velocity} = \frac{\text{Displacement}}{\text{Time}} = \frac{2 \, \text{m}}{\pi \, \text{s}} \]

### Step 4: Calculate the Average Acceleration
Average acceleration is defined as the change in velocity divided by the time taken.

The initial velocity \( \vec{v_i} \) is directed tangentially at the starting point, and the final velocity \( \vec{v_f} \) is directed tangentially at the opposite point. Since these velocities are in opposite directions, we can consider their magnitudes:

\[ \text{Change in velocity} = |\vec{v_f} - \vec{v_i}| = |(-1) - (1)| = 2 \, \text{m/s} \]

Now, we can calculate the average acceleration:

\[ \text{Average Acceleration} = \frac{\text{Change in velocity}}{\text{Time}} = \frac{2 \, \text{m/s}}{\pi \, \text{s}} = \frac{2}{\pi} \, \text{m/s}^2 \]

### Summary of Results
1. **Displacement**: 2 m
2. **Distance travelled**: \( \pi \) m
3. **Average velocity**: \( \frac{2}{\pi} \, \text{m/s} \)
4. **Average acceleration**: \( \frac{2}{\pi} \, \text{m/s}^2 \)