Relative Velocity
Frame of Reference
When we've used velocity in the past, we've always assumed that the earth was the reference frame. When we said things like "A car has a velocity of 30m/s in the x-direction", it was always assumed that the coordinates are fixed with respect to earth.
Earth is not the only reference frame from which we observe things. When you are driving in a car, you are in a reference frame that moves with the car. From that reference frame, the car around you, being part of your reference frame, is seen as stationary, but the earth is moving. From this frame, other cars on the road may be moving both with respect to you and with respect to the earth, but those velocities will be measured differently depending on the reference frame of the observer.
There is an important physics principle at work here: So long as we view (or calculate) an event from a frame of reference that has a constant velocity (this is called an inertial reference frame), it should lead to the same conclusions. While this principle is true, working out the physics can often be much easier from one frame than another. It turns out that viewing collisions from the center of mass frame makes them much simpler! This is precisely because the momentum of the system is always zero from the center of mass frame.
Notation
We need to be explicit about what it is that is being measured or observed, and from which frame the observation is being made. The notation we will use to describe the position of object A as observed from frame B will be and the velocity of A observed from frame B is likewise
Two Rules
We now wish to transform a velocity vector measured from one reference frame, to its corrsponding value as measured from another frame. There are exactly two rules that will allow you to do this: The first rule is that if you reverse the point of view - so rather than observing object A from reference frame B, you observe object B from reference frame A - you must negate the vector, or If this doesn't make sense, consider standing on a sidewalk while observing a car traveling eastward. If you imagine now that you are the person in the car observing a person on the sidewalk, which way do you see the person on the sidewalk moving? In other words, assume the car is actually stationary (since that's how it seems to you). The street and everything on it is coming toward you, which is toward the west. Do you see the reversal of direction? That's all negating the vector does.
The second rule will allow you to transform the frame of reference, and goes like this:
Notice that the center indices match on the right side of the equation (B) and that the first index on the left matches the first index on the first term on the right and the last matches the last. If you stick to these two rules, it will be hard to go wrong when transforming coordinates.
EXAMPLE: Suppose a baseball (subscript 'b') is flying through the air with a velocity measured with respect to the ground (subscript 'g') of An outfielder (subscript 'o') is running toward the ball to catch it at a velocity measured with respect to the ground of What is the velocity of the ball with respect to the outfielder?
SOLUTION: We are looking for We have the first of those vectors, but the second has inverted subscripts, so we must negate the vector to flip the frame of reference. This means Adding the two vectors gives
Limitations
As I've often done in the past, I will mention now that this second equation has limitations. It is actually not valid for large velocities approaching the speed of light. Such situations which would be called the relativistic addition of velocities require a modified version of this equation to satisfy what we know of relativity. That topic belongs in a third semester course including relativity. One interesting result with relativistic velocity addition is that the magnitude of the relative velocity cannot exceed the speed of light, c. That holds true even for cases where light approaches light from opposite directions. The nature of reality is c+c=c in such cases!