A fly crawls with constant speed $v$ along the radial spoke of a wheel which is rotating with constant angular velocity $ω \hat{k}$ . Initially the fly starts from centre of the wheel along $x -$ axis. $\sum$ is fixed frame attached with the axis. ${\hat{e}}_{r}$ is unit vector along radius and ${\hat{e}}_{θ}$ is unit vector along tangent as shown in the figure. Then :

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The velocity of the fly in fixed frame $\sum$ is $\vec{v} = v ω {\hat{e}}_{r} + v {\hat{e}}_{θ}$

The position of fly at any instant is $\hat{r} = v t {\vec{e}}_{r} + ω r t {\hat{e}}_{θ}$

The position of fly at any instant from $O$ is $\vec{r} = \frac{v θ}{ω} {\hat{e}}_{r}$

The velocity of the fly in fixed frame $\sum$ is $\vec{v} = v {\hat{e}}_{r} + r ω {\hat{e}}_{θ}$

answer is A, C.

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Detailed Solution

$r = (v t) {\hat{e}}_{r}$

Also $θ = ω t$

So, $r = (\frac{v θ}{ω}) {\hat{e}}_{r}$

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