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(A) To ensure the water drops enter the tube without touching its sides,the tube must be aligned with the direction of the relative velocity of the ra

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$	an ^{-1}\left(\frac{v}{u}ight)$

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(A) To ensure the water drops enter the tube without touching its sides,the tube must be aligned with the direction of the relative velocity of the rain with respect to the boy.Let the velocity of the boy be $\vec{v}_b = v \hat{i}$ and the velocity of the rain be $\vec{v}_r = -u \hat{j}$.The relative velocity of the rain with respect to the boy is $\vec{v}_{rb} = \vec{v}_r - \vec{v}_b = -u \hat{j} - v \hat{i}$.The angle $	heta$ that the resultant velocity vector makes with the vertical is given by $	an 	heta = \frac{|v_x|}{|v_y|} = \frac{v}{u}$.Therefore,the angle with the vertical is $	heta = 	an ^{-1}\left(\frac{v}{u}ight)$.

$A$ boy is running on a plane road with velocity $v$ with a long hollow tube in his hand. The water is falling vertically downwards with velocity $u$. At what angle to the vertical,he must incline the tube so that the water drops enter it without touching its sides?

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