## Varieties of Collisions
One of the treasured concepts for physicists is the idea of the **conservation of some quantity**. We'll make use of that idea over and over.
While we have a sophisticated justification for this affection for conserved things, even before Conservation Laws were a notion at all, natural scientists had an intuitive sense that nature seemed to preserve some qualities. The first such serious assertion became known as the "Conservation of Momentum." Descartes started it all when he declared that the total “amount of motion” is unchanged, just shared among all of the various bodies in the cosmos.
His model of the universe assumed that it was originally kick-started with all of the material bits set into initial motion and all of this primordial motion shared among all objects forever—a little given up over here means that it shows up over there. The total is preserved, or "conserved." Add those individual bits of motion up at any time and you get the amount of motion you started with. “Bits of motion” for him meant: speed.
This eventually led him to his Big Idea that outer space was (had to be?) filled with various sized balls which were originally rotating together in a great “vortex,” and in that way dragging and pushing the planets along with them. Think of a ball pit for kids (fun, but a veritable petri dish of bacteria.) Those balls constituted his choice as material cause of the orbits of the planets.
> **Wait.** What balls?
> **Glad you asked.** You mean how did he come up with this idea? As a materialist he felt that he was forced to invent a cause: some contact force between objects to make anything move, including the planets. Descartes' philosophy influenced his science: it was top-down. Postulate a cause and then work it out. But don't ever postulate a kind of motion without first setting up the mechanism that creates it. His commitment meant: I see the planets moving, so something has to be pushing them. Material, but invisible balls sharing the original primordial motion is as good as anything else.
Newton dismantled Descartes' vortices but after considerable massaging the preservation of the total motion was an idea that was still around after Newton was finished. Certainly, we now attribute planetary motion about the Sun as reflecting the original rotation of the random space-dust that nearly five billion years ago slowly coalesced into their solid masses. Still orbiting, after all these years.
Let’s look at some collisions that we all could create using everyday objects. I'm aiming toward a symbolic and pictorial approach with only minimal mathematics.
> **Wait.** Apples?
> **Glad you asked.** Well, we could collide apples but that would be messy. When we get to quantum mechanics and particle physics you, like I do, will probably have "billiard balls" in your head.
Here's a very familiar collision in three scenes from the top, to the bottom: (a) before, (b) during, and (c) after the collision of two identical, rigid, frictionless balls. Yes, billiard balls is one of ways that we’ll abstract from real-life collisions to ideal ones. Here you go:
Two identical billiard balls are moving towards one another with the same speeds in (a), collide at (b), and recoil from one another at © Notice that they're slightly off-center which is necessary in order to get them to recoil at an angle.
* First, in order for these two balls to change their motions from totally horizontal [(a): before], to obliquely scattering to directions with both horizontal and vertical momenta [(c): after], they need to collide off-center [(b): during]. So obviously, these particular billiard balls have a size. When we deal in quantum particle collisions, the sizes are so tiny that it simply works to treat them as point-like: no size at all.
* Second, by setting up the situation with no friction, we've left the realm of real billiard balls which roll. Rolling is entirely due to friction (think of your car's tires on ice). Our colliding objects could be in outer space, barreling along without friction.
* That they are "rigid" means that when they collide their material does not compress at all. In real materials, there would be some “give” that would vibrate and dissipate as heat in the ball and vibrate the air producing sound. Our ideal collisions are called "elastic," which we'll explore in the next lesson.
* Finally, no two things in our everyday life are absolutely identical. If you’ve studied any philosophy, you might recognize some of Plato’s reasoning. Real billiard balls would be copies of the Ideal Billiard Ball for him. We nod to Plato but we allow absolutely identical objects in our working-ideal world. (But in the quantum world, all like-objects are absolutely identical—every electron is like every other electron—and that will become an interesting chapter in our QS&BB quantum mechanical story!)
> **Wait.** There you go again. You've decided to describe an unrealistic circumstance. How is that helpful?
> **Glad you asked.** Remember, when we build a model we do it in a way that emphasizes the dominant physics at the expense of any less significant, but realistic effects, that would get in the way of a simple description. In this case it's also a pretty good approximation for many real situations. Likewise if we shot one electron toward another election their electrostatic repulsion is responsible for the recoil and it's almost perfect. So this is a pretty good model for us.
> By the way: our concern in particle physics is the [(b): during] part of this interaction. That’s where the guts of the physics lives and nature’s rules are at work.
> **Wait.** Sorry to bother you again so quickly. Isn’t the “during” here just that the two balls touch and recoil?
> **Glad you asked.** No problem. That’s the neat thing about, well, everything. There isn’t a rule or a force of nature called “touching.” The rule of nature at work here is the same one that keeps you from falling through the floor: the electromagnetic repulsion of the electrons near the surface of each of the balls is the actual force at work in “during.” One of only four known forces. But stay tuned.
This ideal two dimensional collision is mathematically more complicated than we'll need. I wanted to start with something familiar. Let's categorize our collisions by writing little reactions and drawing diagrams to describe them.