A particle is moving on a circular path of radius 10m with uniform speed of `4ms^(-1)`.The magnitude of change in velocity of particle when it completes a semi circular path is

Options

A. `4ms^(-1)`
B. `8ms^(-1)`
C. `10ms^(-1)`
D. zero

Answer

To find the magnitude of the change in velocity of a particle moving in a circular path after completing a semi-circular path, we can follow these steps:

### Step 1: Understand the Initial and Final Velocity Vectors
- The particle is moving with a uniform speed of \(4 \, \text{m/s}\) along a circular path of radius \(10 \, \text{m}\).
- Let's assume the particle starts at point A (initial position) and moves to point B (final position) after completing a semi-circular path.
- At point A, the velocity vector can be represented as \( \vec{V}_{\text{initial}} = 4 \, \text{m/s} \, \hat{i} \) (assuming it is moving to the right along the x-axis).
- At point B, which is directly opposite point A, the velocity vector will be directed downwards, represented as \( \vec{V}_{\text{final}} = -4 \, \text{m/s} \, \hat{j} \).

### Step 2: Calculate the Change in Velocity
- The change in velocity \( \Delta \vec{V} \) can be calculated using the formula:
\[ \Delta \vec{V} = \vec{V}_{\text{final}} - \vec{V}_{\text{initial}} \]
- Substituting the values:
\[ \Delta \vec{V} = (-4 \, \hat{j}) - (4 \, \hat{i}) = -4 \, \hat{i} - 4 \, \hat{j} \]

### Step 3: Find the Magnitude of the Change in Velocity
- The magnitude of the change in velocity can be found using the formula:
\[ |\Delta \vec{V}| = \sqrt{(-4)^2 + (-4)^2} \]
- Calculating this gives:
\[ |\Delta \vec{V}| = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \, \text{m/s} \]

### Conclusion
- Therefore, the magnitude of the change in velocity of the particle when it completes a semi-circular path is \( 4\sqrt{2} \, \text{m/s} \).