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)

$\frac{\pi}{2 \sqrt{2}}$

Answer 2

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, $\frac{v_{1}}{v_{2}}=\frac{\pi}{2 \sqrt{2}}$