Answer
$V_{B/C} = -39\frac{ft}{s}$
Work Step by Step
Only one line is present in this problem. Summate each respective segment to obtain the total length of moving rope:
(1) $l = S_{A}+2\times S_{B}+2\times S_{C}$
Take the derivative of equation (1) to obtain the respective velocities of components A, B, and C:
(2) $0 = V_{A}+2\times V_{B}+2\times V_{C}$
The problem tells us that $V_{A} = 6\frac{ft}{s}$ downwards and $V_{C} = 18\frac{ft}{s}$ downards. Plug these values into eqation (2) to calculate $V_{C}$:
$0 = 6+2\times V_{B}+2\times 18$
$0 = 42 + 2\times V_{B}$
$2\times V_{B} = -42$
$V_{B} = -21\frac{ft}{s}$
Since we treated downward velocity as positive, a negative velocity indicates that rope segment B is moving upwards. Note that $V_{B} = -21\frac{ft}{s}$ is not the final answer. The question wants the relative velocity of $V_{B}$ in terms of $V_{C}$:
(3) $V_{B/C} = -21 - 18 = -39\frac{ft}{s}$
$V_{C}$ takes the same sign as $V_{B}$ since rope segment C is moving in the opposite direction of $V_{B}$.
