Relative Velocity Help I am struggling with relative velocity and so decided to do some extra problems in my textbook. I have 2 problems that seem to contradict, and its annoying things like this that throws me off, leaving me unable to solve problems due to simple concepts of relativity. This is very frustrating. Anyway, my question: 
In this first question : 

The 4m/s velocity to the left of the box is apparently relative to the boat. OK, let's assume that I believe this, which I don't see why at all.
Let's look at the 2nd problem :

The velocity of the box should be relative to the cart, just like the first question. The spring "pushes" the box giving it a velocity RELATIVE to the cart, but no, its not this in the solutions. Why is it different?
As I asked in a question earlier, I don't get relative velocities (its been a year, I have read several resources, I still don't get it).
When is a velocity absolute (fixed reference frame), and when is a velocity relative (moving reference frame)?
THIS IS NOT HOMEWORK
 A: Good question.  Your confusion is very understandable.  But if you're careful, you'll see where your thinking is going wrong.
These two questions are saying two different things.  The first question tells you what the velocity is, and it tells you that that velocity is measured relative to the boat.  You can measure a velocity relative to anything you want.  In fact, right there in the solution to the first question, they calculate the velocity relative to the water (the stationary inertial coordinate system).
Now the second question comes along.  It never tells you the velocity of the box relative to the cart; it's asking you to calculate it.  And that's what the solution does.  Both of these solutions actually work in the stationary reference frame.  But the first question gave you the velocity relative to the boat and asked you to solve for the velocity relative to the water, while in the second question you had to solve for the velocity relative to the ground, and then calculate the velocity relative to the cart.
So any time you talk about a velocity, you need to also talk about what reference frame (coordinate system) that velocity is expressed in.  If you know the velocity in one reference frame, and know the relative motion of any other reference frame, you can calculate the velocity in that second reference frame.
