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(A) Let the velocity of block $B$ be $V_B = 3 	ext{ m/s}$ (towards left) and the velocity of block $A$ be $V_A = 1 	ext{ m/s}$ (towards left).Block

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$5$

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(A) Let the velocity of block $B$ be $V_B = 3 	ext{ m/s}$ (towards left) and the velocity of block $A$ be $V_A = 1 	ext{ m/s}$ (towards left).Block $C$ is constrained to move vertically along the face of block $B$. Let $V_{C/B}$ be the velocity of block $C$ with respect to block $B$.Considering the string constraint,the length of the string remains constant. The pulley attached to block $A$ moves with velocity $V_A$. The string segment between block $A$ and the fixed pulley on $B$ has a length $L$. The velocity of the pulley on $A$ is $V_A$. The velocity of the string segment moving towards $B$ is related to the relative motion.By the constraint equation for the pulley system,the vertical velocity of block $C$ relative to block $B$ is given by $V_{C/B} = 2(V_B - V_A) = 2(3 - 1) = 4 	ext{ m/s}$ (downwards).Since block $C$ is in contact with block $B$,it also moves horizontally with the velocity of block $B$,which is $V_{C,x} = 3 	ext{ m/s}$ (towards left).Thus,the velocity vector of block $C$ is $\vec{V}_C = -3 \hat{i} - 4 \hat{j} 	ext{ m/s}$.The magnitude of the velocity is $V_C = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 	ext{ m/s}$.

The velocities of block $A$ and block $B$ are shown in the figure. The velocity of block $C$ is ......... $m/s$ (assume that the pulleys are ideal and the string is inextensible).

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