A particle moves along a circle with a uniform speed v. After the position vector has made an angle of `30^@` with the reference position, its speed will be

Options

A. `vsqrt2`
B. `v/sqrt2`
C. `v/sqrt3`
D. v

Answer

To solve the problem step-by-step, let's break down the concepts involved in uniform circular motion and the specific question regarding the speed of the particle after it has moved to a position where its position vector makes an angle of 30 degrees with the reference position.

### Step 1: Understand the Concept of Uniform Circular Motion
In uniform circular motion, a particle moves in a circular path with a constant speed. This means that while the speed (magnitude of velocity) remains constant, the direction of the velocity vector changes continuously as the particle moves along the circular path.

**Hint:** Remember that in uniform circular motion, the speed is constant, but the velocity (which includes direction) changes.

### Step 2: Identify the Given Information
- The particle moves with a uniform speed \( v \).
- The position vector has made an angle of \( 30^\circ \) with the reference position.

**Hint:** Identify what is being asked: the speed of the particle, not the velocity.

### Step 3: Analyze the Speed at the New Position
Since the particle is moving with a uniform speed, regardless of its position on the circular path, the speed remains constant. Thus, even after the position vector has made an angle of \( 30^\circ \), the speed of the particle is still \( v \).

**Hint:** Focus on the definition of speed in uniform circular motion; it does not change with position.

### Step 4: Conclusion
The speed of the particle after it has moved to the position where the position vector makes an angle of \( 30^\circ \) with the reference position is still \( v \).

**Final Answer:** The speed of the particle is \( v \).