Question

A pipe which can rotate in a vertical plane is mounted on a cart. The cart moves uniformly along a horizontal path with a speed $v_1=2m/s$. At what angle $\alpha$ to the horizontal should the pipe be placed so that drops of rain falling with a velocity $v_2=6m/s$ move parallel to the walls of the pipe without touching them? consider the velocity of the drops as constant due to the resistance of the air

• A. $ta{n}^{-1}\left(3\right)$
• B. $ta{n}^{-1}\left(\frac{1}{3}\right)$
• C. $ta{n}^{-1}\left(\frac{1}{2}\right)$
• D. None of these

Answer 1

The correct option is A $ta{n}^{-1}\left(3\right)$

Given, speed of cart $V_P=v_1=2m/s$
And, speed of rain drops $V_R=v_2=6m/s$
Drops of rain move parallel to the walls of pipe if $V_{rp}$ makes $\alpha$ anlge with the horizontal.


$\therefore$ $\overrightarrow{V_{RP}}=\overrightarrow{V_R}-\overrightarrow{V_P}$
$=\overrightarrow{V_R}\hat{j}-(-\overrightarrow{V_C})\hat{i}$
$\Rightarrow v_2\hat{j}+v_1\hat{i}$
And, we have
$tan\alpha =\frac{v_2}{v_1}=\frac{6}{2}$
$\alpha =ta{n}^{-1}\left(3\right)$