The mass-momentum vector

Situations that can be addressed within Newtonian dynamics are cases in which the velocities of objects are much slower than the speed of light c, and the Galilei transformation suffices as the transformation from one inertial frame to another.

Now, we would like to answer the questions posed in the previous section using diagrams. But for that we must be able to represent pictorially what we mean by the term “same impact.” In the current case, the “impact” we are talking about is that which accelerates the baseball from “a state in which it is at rest” to “a state in which it is traveling at +1 meters per second.” But this in turn means that we must first be able to represent pictorially the “state of the baseball moving at velocity υ” for generic velocities υ.

So how can we do this? Of course, the motion of any object is described by its worldline on the spacetime diagram, and the object's velocity is encoded in the slope of the worldline. But there are two reasons why the worldline is not an appropriate representation of the “state of motion” of an object:

1. Depending on whether the object is a baseball, a ping-pong ball, or a bowling ball, the amount of “stuff” that is moving is different, but the worldline does not give you that information.

2. […]