A particle is moving with constant speed in a circular path. When the particle turns by an angle 90 ° , the ratio of instantaneous velocity to its average velocity is π : x 2 . The value of x will be

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In circular motion with constant speed, magnitude of instantaneous velocity, $v_{i n s} = ω r$ .
Time taken to turn by $90 °$ will be $= \frac{\frac{π}{2}}{ω} = \frac{π}{2 ω}$ .
Displacement will be equal to the length of $A B = \sqrt{r^{2} + r^{2}} = r \sqrt{2}$
Average velocity
$v_{a v g} = \frac{displacement}{time} = \frac{r \sqrt{2} \times 2 ω}{π} = \frac{2 \sqrt{2}}{π} ω r$
Hence, required ratio $\frac{v_{ins}}{v_{avg}} = \frac{ω r π}{2 \sqrt{2} ω r} = \frac{π}{2 \sqrt{2}}$
Therefore, $x = 2$ .

Physics - Circular Motion Question with Solution | TestHub

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