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• 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 meansure, 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 movingOne stops and the other starts

• One object twenty times as big and not movingThe 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, unaffectedWhich 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 dimensionAnd, 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  about out Brilliant, this video's sponsor.

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

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# Most Collisions Are Secretly in One Dimension

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Summer posted on 2022/03/03
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