A block is dragged on a smooth plane with the help of a rope which moves with a velocity  $v$ as shown in figure. The horizontal velocity of the block is 

Let the string is coming out from the pulley at a rate of $v^1$m/s.

The component of this velocity will be the horizontal velocity of the block moving on the surface.

${\text{ν}}^1\cos ({90}^o-\theta )=\nu \Rightarrow {\text{\hspace{0.17em}ν}}^1=\frac{\text{ν}}{\sin \theta }$