When two things crash into each other, it seems like it should be a messy affair,
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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