DISCUSSION
Since the problem asks for a relationship between the change of speeds of A and B, and the distance traveled by B, the work/energy equation is a natural method of choice. Recall that with the work/energy equation, we typically want to include as much in the system in order so that we can make as many forces to be workless, internal forces within the system. To this end, consider a system made up of A, B, the cable and the pulleys.

Hints:
You should follow the four-step solution plan described in the lecture book:
Step 1: Free body diagram (FBD) - Draw an FBD of the system made up of A+B+cable+pulleys. Which forces in your FBD do work on the system?
Step 2: Kinetics - Write down the work/energy equation for the system and the terms included in this equation.
Step 3: Kinetics - Recall our work back in Chapter 1 when we related the speeds of two blocks connected by a cable/pulley system.
Step 4: Solve - Solve for the final speeds of A and B.

Ask and answer questions below. You will learn from both asking and answering.