# Step-by-Step Solution ## Problem Recap A particle moves in a circle of radius \( r \) with center \( O \) and constant speed v \). It moves from \( A \) to point \( B \), where \( \angle AOB = \theta \). **Question:** What is the change in velocity (\(\Delta \vec{v}\)) as it moves from \( A \) to \( B \)? --- ## Step 1: Understanding the Problem - The velocity at any point is tangent to the circle. - At \( A \) and \( B \), the magnitudes of velocity are equal (\( v \)), but the directions are different. - We are asked for the change in the velocity vector (i.e., \(\vec{v}_B - \vec{v}_A\)), **not** the change in speed. --- ## Step 2: Drawing the Velocity Vectors Let’s assume: - The velocity at \( A \) is tangent to the circle at \( A \). - The velocity at \( B \) is tangent to the circle at \( B \). - Both vectors have magnitude \( v \). - The angle between these two velocity vectors is exactly equal to the angle at the center: \( \theta \). --- ## Step 3: Using Vector Subtraction The change in velocity vector is: \[ \Delta \vec{v} = \vec{v}_B - \vec{v}_A \] The magnitude of the change is given by the chord length in the velocity space: \[ |\Delta \vec{v}| = \sqrt{v^2 + v^2 - 2vv\cos\theta} \] \[ |\Delta \vec{v}| = \sqrt{2v^2(1 - \cos\theta)} \] \[ |\Delta \vec{v}| = v\sqrt{2(1 - \cos\theta)} \] --- ## Step 4: Matching with the Answers Let’s simplify \( v\sqrt{2(1 - \cos\theta)} \): \[ 1 - \cos\theta = 2\sin^2\left(\frac{\theta}{2}\right) \] So, \[ |\Delta \vec{v}| = v\sqrt{2 \cdot 2\sin^2\left(\frac{\theta}{2}\right)} = v\sqrt{4\sin^2\left(\frac{\theta}{2}\right)} = 2v\sin\left(\frac{\theta}{2}\right) \] None of the given options match this value. --- ## Step 5: **Final Answer** ### **Correct Answer:** **D) None of these** **Explanation:** The actual change in velocity is \( 2v\sin\left(\frac{\theta}{2}\right) \), which is not present among the options. --- ### **Summary Table** | Step | Calculation/Reasoning | Result | |------|----------------------------------------|-----------------------------| | 1 | Velocity vectors at \( A \) and \( B \)| Separated by angle \( \theta \) | | 2 | Magnitude of change in velocity | \( v\sqrt{2(1-\cos\theta)} \) | | 3 | Trig simplification | \( 2v\sin(\theta/2) \) | | 4 | Matching with options | **Not present** | | 5 | **Correct Option** | **D) None of these** |