Question

A body is moving with a constant speed \(v\) in a circular path of radius \(r\). The magnitude of average velocity after half a revolution is: 

• A. \(\frac{2v}{\pi}\)
• B. \(v\)
• C. \(\frac{v}{\pi}\)
• D. zero Correct Answer: (A) \(\frac{2v}{\pi}\) Solution:

Hint:

In circular motion, average speed depends on total distance travelled, whereas average velocity depends only on displacement. After half a revolution, displacement equals the diameter of the circle.

Answer 1

The Correct Option is A

Concept: Average velocity is defined as the ratio of total displacement to total time taken. It is a vector quantity and depends on the net displacement, not on the total distance travelled. \[ \text{Average Velocity}=\frac{\text{Displacement}}{\text{Time}} \] For circular motion, displacement is the shortest straight-line distance between the initial and final positions.

Step 1: Determine the displacement after half a revolution. When the body completes half a revolution, it reaches the diametrically opposite point of the circle. Hence, the displacement is equal to the diameter: \[ \text{Displacement}=2r \]

Step 2: Calculate the time taken for half a revolution. The distance travelled during half a revolution is the semicircular arc: \[ \text{Distance}=\pi r \] Since speed is constant and equal to \(v\), \[ \text{Time}=\frac{\pi r}{v} \]

Step 3: Find the average velocity. \[ \text{Average Velocity} =\frac{2r}{\pi r/v} =\frac{2rv}{\pi r} =\frac{2v}{\pi} \] Therefore, \[ \boxed{\text{Average Velocity}=\frac{2v}{\pi}} \] Hence, the correct option is \(\boxed{(A)}\).