## Ideal Collisions You have all collided one thing with another, whether in a game (on purpose) or your car (not on purpose!)—one thing banging with another thing is a well-known, and sometimes personal event. I’ll refer to collisions many times and in this section we’ll collect a set of representative kinds of collisions for future reference. ### Collision 1: $\mathbf{A}+\mathbf{B}\to \mathbf{A}+\mathbf{B}$ >**Seriously:**

>Please write the above "formula" and draw the corresponding picture! Let's start simply with two billiard balls which are in a head-on collision, that is, *not* off-center. For this first collision, the objects are identical but they’re labeled separately as $A$ and $B$ in order to keep track of them. At the LHC, they would be identical protons. In billiards they would be different colored balls. We can describe these or any collisions of two equal particles in, two equal particles out, with the same “reaction” formula: $\mathbf{A}+\mathbf{B}\to \mathbf{A}+\mathbf{B}$ (or maybe you’d prefer $\mathbf{A}+\mathbf{A}\to \mathbf{A}+\mathbf{A}$?).
TIME GOES FROM TOP TO BOTTOM IN THREE STEPS: Collision 1: $\\mathbf{A}+\\mathbf{B}\\to \\mathbf{A}+\\mathbf{B}$
In Collision 1 the two balls retain their identities after they collide: you (a) start with an $A$ and $B$ at the beginning and you (c) end up with those same pair of $A$ and $B$ at the end, none the worse for wear. In (b) they collide and if they’re everyday objects, you could take a picture and see them touch. I’ll bet you know what would happen if they are identical in mass and their initial speeds, $v$, are the same. Descartes would have said that the total speed at before they collide is $2v$ and so the total speed after they collide would also be $2v$. And he would be sort of right—but for the wrong reason (most of us hate it when that happens). In this collision, both balls would recoil from one another and just reverse their original motions. But this a special case since one could be faster than the other in (a) or one could even be sitting still: at rest. Then their final speeds will be different and Mr Newton and Mr Huygens wrote down the rules that enable us to predict the results: you give me their states at the beginning (remember, a state means knowing positions and momenta) and I can tell you their states at the end. The middle is where the physics action is since a collision like this could be two actual electrons which would repel one another without ever touching. Stay tuned. ### Collision 2: $\mathbf{a}+\mathbf{B}\to \mathbf{a}+\mathbf{B}$ >**Seriously:**

>Please write the above "formula" and draw the corresponding picture! A variation on Collision 1 is when the two participants have different masses or different natures altogether. The reaction formula looks similar, but I’ll call out the lighter one with obvious notation: $\mathbf{a}+\mathbf{B}\to \mathbf{a}+\mathbf{B}$.
TIME GOES FROM TOP TO BOTTOM IN THREE STEPS: Collision 2: $\\mathbf{a}+\\mathbf{B}\\to \\mathbf{a}+\\mathbf{B}$
Think a 180 lb safety, stiff-armed and bouncing off a 250 lb running back. Those collisions are not symmetric and are heavily dependent on the initial speeds and masses. Collision 1 is really a special case of Collision 2, which is the most general everyday (ideal) collision between two, different objects. ### Collision 3: $\mathbf{A}+\mathbf{B}\to \mathbf{C}$ >**Seriously:**

>Please write the above "formula" and draw the corresponding picture! If two of you are really good, you could add glue to a couple of billiard balls and precisely throw them at one another with exactly the same speeds. They would stick and stop dead, becoming one object (C) made up of the original two. (We’ll see why in a moment.) > **Wait.** That doesn’t sound like a good idea.

> **Glad you asked.** I agree. But these are ideal collisions, so let’s pretend that you two are ideal throwers. Here’s the picture (see the glue?):
TIME GOES FROM TOP TO BOTTOM IN THREE STEPS: Collision 3 with glue:$ \\mathbf{A}+\\mathbf{B}\\to \\mathbf{C}$ The glue is brown.
Here, Descartes got it wrong: he would have said that the original motion was $2v$, but we know that the final “motion” is $0$. Obviously he didn't appreciate the importance of the *directions* of the *velocities* in the addition to their magnitudes. That is, he didn't know about vectors. In the sticking-together collision we instinctively know that the result is: $v_A-v_B = 0$ so that they stop. ### Speed Is Not the Thing: Momentum is the Thing *Speed* is not the conserved quantity. Rather, Newton's *momentum* was the key, defined as the vector quantity, $\vec{p} = m\vec{v}$, with both a magnitude and direction. He had the beginnings of an appreciation for the direction of momentum, filling a concept-gap that neither Descartes nor Galileo had come to on their own.
A woodcut from Huygens’ work illustrating his little 17th century particle accelerator. One person with a pendulum is in a boat on an Amsterdam canal and the other is on the pavement with a pendulum as well. Back and forth they went, with different masses dangling from their ropes. It must have been a sight.
Huygens got it right: he understood the idea of vectors, although in an entirely different and complicated way that got sorted out by the 19th century into the language that we use today. Collisions were a fascinating study for those working in the 17th century. Everyone understood that friction confused the real picture, so people relied on colliding pendulums where these effects were reduced. One has visions of everyone having many "executive toy" contraptions in their workrooms, changing out the bobs and causing clacking collisions with careful measurements of the outcomes. Huygens made use of his home-town canals in Amsterdam as way to collide masses in a controlled way. Amsterdam's canals provided a near-frictionless racetrack for accelerating particles. He would station a colleague on a canal-boat with a stationary pendulum which would collide with one held by someone on the shore. As the boat went by, nearly frictionless collisions were created and, he was able to get study the collision from the point of view of a “fixed” coordinate system (say, the shore-guy when the boat-guy went by) and the “moving” coordinate system (the boat-guy). His geometrical explanation was very complicated, but essentially correct. I'll describe it here in modern language. Back to Collision 3 with the glue: if $A$ and $B$ are different masses, but they still stick, they would still make a third object (C). The idea is the same: two objects become one object. That’s the situation of our safety and running back where the safety hangs on and really tackles the runner. Or, where a baseball is caught by a fielder, again, two objects becoming one. Together, they’re a third object. Collisions 1-3 are all of the collisions that can happen in one dimension with two everyday objects. But this is QS&BB, so there’s more!