In the given figure the velocity and acceleration of B are $v_B,a_B$  if instantaneous velocity and acceleration of A are as shown in the Fig. Find  $\frac{a_B}{v_B}$

 

We use the fact that string is inextensible and length of string is constant
$l_1+l_2+l_3+l_4=$  constant differentiate to get  $\frac{d{l}_1}{dt}+\frac{d{l}_2}{dt}+\frac{d{l}_3}{dt}+\frac{d{l}_4}{dt}=0$
$v_A+v_A+(-v_C)+(-v_C)=0$  
 
Or  $v_c=v_A\Rightarrow v_C=1m/s\uparrow$
Where $v_C$  is velocity of pulley C. 
Similarly  $a_C=a_A$
 $\therefore a_C=2m/s^2\downarrow$
Now  $l_4+l_5=l_6$
 $\therefore \frac{d{l}_4}{dt}+\frac{d{l}_5}{dt}=\frac{d{l}_6}{dt}$
 $\therefore -v_C+\left(v_B\right)=-v_B$
  $\therefore v_B=0.5m/s\uparrow$
Similarly,  $a_B=\frac{a_C}{2}\therefore a_B=1m/s^2\downarrow$