Question

A particle is moving with a constant speed in a circular path. The ratio of average velocity to its instantaneous velocity when the particle describe an angle $\frac{\pi }{2}$ is?

• A. $3/2\pi$
• B. $\frac{2\sqrt{2}}{\pi }$
• C. $\frac{2}{\pi }$
• D. $\frac{1}{\pi }$

Answer 1

The correct option is B $\frac{2\sqrt{2}}{\pi }$

When the particle describe an angle $\frac{\pi }{2}$, then distance $\sqrt{R^2+R^2}=\sqrt{2}R$

Time $=\frac{T}{4}$

$\therefore v_{avg}=\frac{\text{Distance travel}}{\text{Time required}}=\frac{\sqrt{2}R}{T/4}=\frac{4\sqrt{2}R}{T}$

$\omega =\frac{2\pi }{T}\Rightarrow T=\frac{2\pi }{\omega }$

$v_{avg}=\frac{4\sqrt{2}g}{\frac{2\pi }{\omega }}=\frac{4\sqrt{2}R}{1}\times \frac{\omega }{2\pi }=\frac{2\sqrt{2}R\omega }{\pi }$

Also, $v_{ins}=R\omega$

$\therefore \frac{v_{avg}}{v_{ins}}=\frac{\frac{2\sqrt{2}R\omega }{\pi }}{R\omega }=\frac{2\sqrt{2}}{\pi }$.