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 curious, and so you may also be curious
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