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

Let at any instant of time, the length AB be l, here angle $\mathrm{\theta }$ and length l vary with time, then using Pythagoras theorem in $∆ABC$,

$x^2 + y^2 = t^2$

On differentiating both sides w.r.t t, we get
or  $2x\frac{dx}{dt}+2y\frac{dy}{dt}=2l\frac{dl}{dt}$

As, there is no vertical motion of the block, so

$\frac{dy}{dt}=0,\frac{dx}{dt}=v_x\text{ and } \frac{dl}{dt}=v$

$\therefore 2{\mathrm{xv}}_{\mathrm{x}} = 2\mathrm{lv}$

$\text{ or } v_x=\frac{l}{x}v \text{ or } v_x=\frac{v}{\left(\frac{x}{l}\right)}=\frac{v}{\sin \theta }$