A block is pulled on a smooth surface with the help of a rope and ropc is pulled with speed $u$ as in figure. Find the horizontal velocity of the block. (Assume the contact docsn't losc during motion.)
Solution
$x+c+\sqrt{y^{2}+d^{2}}-$ constant
$\Rightarrow \frac{d x}{d t}+\frac{d \sqrt{y^{2} \mid d^{2}}}{d t}-0$
$\Rightarrow \quad u+\frac{1}{2 \sqrt{y^{2} \mid d^{2}}} \times 2 y \frac{d y}{d t}-0$
$\Rightarrow u+\frac{y}{\sqrt{y^{2} \mid d^{2}}}\left(\frac{d y}{d t}\right)-0 \Rightarrow u+\sin \theta(-v)-0$ as $\left(\frac{d y}{d t}\right)--v$
$\Rightarrow \sin \theta(v)-u \quad \therefore \frac{v}{u}-\operatorname{cosec} \theta$