Question

If block B moves to the right with constant velocity 10 m/s, then velocity of Block A relative to B is

• A. 5 m/s left
• B. 5 m/s Right
• C. 15 m/s left
• D. 15 m/s Right

Answer 1

The correct option is B 5 m/s Right


Choose the origin $O$ at a suitable point. Let the coordinates of the pulleys and the fixed point wrt. the origin be $x_A,x_B,$ and $x_C$.
The length $l$ of the string is then
$$l=(x_B-x_F)+2(x_B-x_A)=3x_B-x_F-2x_A$$ So no matter how the two blocks move, the length $l$ of the string remains constant.
Now find the time derivative of the above equation noting that $l$ and $x_F$ are constants.
We get
$\begin{array}{cc}0=& 3\frac{d{x}_B}{dt}-0-2\frac{d{x}_A}{dt}\\ & \Rightarrow 3v_B=2v_A\\ & \Rightarrow v_A=\frac{3}{2}{v}_B\end{array}$
So relative velocity of A wrt. B is
$\Rightarrow v_{AB}=v_A-v_B=\frac{3}{2}{v}_B-v_B=\frac{1}{2}{v}_B=\frac{1}{2}10=5m/s$ Positive, so towards right