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

• A. $\frac{\pi }{2}$
• B. $\frac{3}{\pi }$
• C. $\frac{2}{\sqrt{3}\pi }$
• D. $\frac{\pi }{2\sqrt{2}}$

Answer 1

Answer: (D)

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, $\frac{v_1}{v_2}=\frac{\pi }{2\sqrt{2}}$