In figure, lower pulley is free to move in vertical direction only. Block A is given a uniform velocity u as shown, what is velocity of block B as a function of angle θ.

Let in a small time dt, displacement of A is dx and displacement of B is dy. Increase in length l1= dx.Increase in length l2=dxsin⁡θ−dycos⁡θBut net increase in length should be zero, sodx+dxsin⁡θ−dycos⁡θ=0⇒ dy=(1+sin⁡θ)dxcos⁡θ⇒ dydt=1+sin⁡θcos⁡θdxdt⇒v=1+sin⁡θcos⁡θu