where just about anything can happen.  I mean, that's kind of our everyday  

experience of collisions! But there's actually  a magic simplicity underlying the complexity. 

In fact, if you have just two things  colliding along just one direction,  

then there's only one possible outcome! I mean,  sure, after the collision each object could in  

principle have any possible velocity to the left  or right - which is to say, there are two unknown  

variables. But conservation of momentum provides  one equation those variables have to satisfy.  

And conservation of energy provides another  equation those variables have to satisfy.  

And in our universe, two independent equations  for two unknown variables will uniquely  

determine those variables. So for each possible  combination of masses and incoming velocities,  

there's only one possible outcome  of a 1D collision. For example. 

Two identical objects coming in at the  same velocity? They bounce off each other. 

One of those objects not moving?  One stops and the other starts. 

One object twenty times as big and not moving?  The little one bounces back with 90% the speed,  

and the big one starts moving with 10% the speed. And so on... 

Oh, “but what if energy isn't conserved?”  Well, yeah, maybe some of the energy of  

the colliding objects doesn't stay as  kinetic energy but turns into heat,  

or sound, or rotational energy, or whatever, so  the conservation of energy equation isn't valid.  

Except, you can simply put the lost energy  into the conservation of energy equation and  

it becomes valid again. So there are still two  equations and two unknowns, and therefore only  

one possible outcome of the collision as far as  the objects' velocities are concerned. Though  

it's typically really hard to keep track of lost  energy and so the outcome of these collisions can  

seem surprising - but from the Universe's  perspective, they are uniquely determined. 

And what about in two or three dimensions where  most collisions aren't perfectly one dimensional?  

Well, the truth is, they secretly are! Most of  the time, collisions in 2D or 3D result in a  

net force between the objects which is only in one  direction - typically perpendicular to the surface  

where the objects collide, though if the surface  is complicated or there's friction it might be a  

different direction. Since there are no net forces  in directions perpendicular to the net force,  

the motion of the objects in those perpendicular  directions is unaffected by the collision!  

So even though a collision happens in 2D, if  you find the right direction the collision  

will be the same as a one dimensional collision  in that direction, and in the other direction,  

the objects just pass by each other, unaffected.  Which means that even in two or three dimensions,  

once you know the secret direction, the outcome  of collisions is again uniquely determined! 

And that's the magic of collisions: even  though they look complicated and random,  

they're secretly not. The combination of  conservation of momentum and conservation  

of energy and the fact that most collisions are  secretly in one dimension means that the outcome  

of almost any collision between two objects is  completely determined - as long as you know the  

incoming masses and velocities, the amount of  kinetic energy lost to heat and sound and so on,  

and the direction of the one secret dimension.  And, as long as you're ignoring quantum mechanics. 

Since most big and complicated collisions are  actually made up of lots of two-object collisions,  

that means big complicated collisions  are also completely determined!  

Which is why it's really easy for  computers to simulate lots of collisions. 

If you've made it this far into a video about  the physics of collisions, I bet you're pretty  

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