Question:

A ball moves one-fourth $\left(\frac{1^{\text {th }}}{4}\right)$ of a circle of radius $R$ in time $T$. Let $v_{1}$ and $v_{2}$ be the magnitudes of mean speed and mean velocity vector. The ratio $\frac{v_{1}}{v_{2}}$ will be

Updated On: Jun 20, 2022
  • $\frac{\pi}{2}$
  • $\frac{3}{\pi}$
  • $\frac{2}{\sqrt{3} \pi}$
  • $\frac{\pi}{2 \sqrt{2}}$
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The Correct Option is D

Solution and Explanation

Mean speed of a moving body,
$v_{ rms }=\frac{\text { total distance }}{\text { total time taken }}$
Mean velocity vector for a moving body,
$v_{ vv }=\frac{\text { total displacement }}{\text { total time taken }}$
A ball moving in a circular arc is shown in the figure below,



Here, $v_{ rms }=\frac{2 \pi R}{4} \times \frac{1}{T}=v_{1}$
Similarly, $v_{ vv }=\frac{\sqrt{2} R}{T}=v_{2}$
Hence, $\frac{v_{1}}{v_{2}}=\frac{\pi}{2 \sqrt{2}}$
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Concepts Used:

System of Particles and Rotational Motion

  1. The system of particles refers to the extended body which is considered a rigid body most of the time for simple or easy understanding. A rigid body is a body with a perfectly definite and unchangeable shape.
  2. The distance between the pair of particles in such a body does not replace or alter. Rotational motion can be described as the motion of a rigid body originates in such a manner that all of its particles move in a circle about an axis with a common angular velocity.
  3. The few common examples of rotational motion are the motion of the blade of a windmill and periodic motion.