WEBVTT
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MARKUS KLUTE: Welcome back
to 8.20, Special Relativity.
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In this section we're
going to talk a bit more
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about collisions.
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I've already seen collisions in
study of momentum conservation
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in previous sections.
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So here we can have a collision.
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Then we can describe them
in the center of mass frame,
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for example, where the total
momentum is equal to 0.
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So in the case of the
collision of two particles,
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the momentum of particle
one plus the momentum
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of particle two is equal to 0.
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We can then describe the
energy and the momentum
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of the particles before
and after the collision.
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In the lab frame, the
situation is different.
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Here typically we
have one particle
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with some momentum hitting
another particle, which
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is at rest.
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But we can also have
different types of collisions.
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We describe or characterize
elastic collisions
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where the kinetic energy is
conserved and so is the mass.
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So you can think about two
billiard balls colliding
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without any friction,
in which case
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they don't change their
appearance, their mass.
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Everything is unchanged, so
you must change the direction.
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The total kinetic energy
in these collisions
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are typically conserved.
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But we can also have
inelastic collisions.
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And there's two different kinds.
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There's sticky kinds, where
the mass after the collision
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is greater.
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So you have two
particles, for example,
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maybe they stick together--
they're some, like,
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Play Dough balls--
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and the kinetic energy after
the collision is smaller.
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Or you can have explosive
collisions, where
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the mass afterwards is smaller.
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Maybe you start from
one heavy, big object
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and then which explodes
into many smaller ones.
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But the kinetic energy after
the collisions is much smaller.
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Those are also collisions.
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So here we want
to do an activity
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and study an
inelastic collision.
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So before we have two particles
there, or billiard balls.
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They're exactly the same
and have a velocity u.
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And after the collision their
mass is capital M, big mass.
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And you're going to
describe this collision once
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in the center of mass frame and
one in the laboratory frame.
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And so the question
now is, are the masses
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and is energy conserved
in those collisions?
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And you're going
to just described
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this in both reference forms.
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So again, stop the video here
and try to work this out.
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I already did this,
so I discussed
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before, in those
collision problems
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it's always important
to be really clear.
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The situation
before the collision
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was A. The situation
after the collision was B.
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So I'm describing this here.
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First in the center of
mass frame where the x--
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and I'm just talking
about x component here--
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the x momentum is 0, which
is equal to the mass times u
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times gamma minus the
mass times u terms gamma.
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That's the 0.
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The energy before is 2 times
the mass times gamma times
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c squared.
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After the collision,
the particle is at rest.
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The new one
particular is at rest
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and has an energy,
large M over--
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times c squared.
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In the laboratory frame
situations, different case.
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X momentum 0 minus
m times u prime--
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this is a different velocity--
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times gamma of u prime.
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So here I'm trying to indicate
that this gamma is not
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the same gamma as over here.
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This is a gamma, but it's
the velocity of u prime.
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And the energy is the rest
mass of the particle addressed
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plus the mass times
gamma times c squared off
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the second particle.
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After the collision, the
particle has some velocity u.
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And so the momentum
in x direction
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is minus large M times u
times gamma of u again.
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And the energy is large M
times gamma u times c squared.
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OK, good.
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So now we can use
momentum conservation
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and find this equation here.
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And from which we
can then calculate
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that the large mass is equal
to 2 times the smaller mass.
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So what you find, and this
is the relativistic math,
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you find that at the conclusion
that the rest mass is not
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conserved.
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The mass of this big
ball is not simply
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the mass of the two
rest masses, or 2 times
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the mass of the rest mass.
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You have to consider
this gamma factor here.
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It's 2 times the relativistic
math, if you want.
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But you also find
that the total energy
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is conserved in circulation so
that the sum of m0 gamma times
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c squared is conserved
in the collision,
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irrespectively in how
you actually reference it
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when you discuss the problem.
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I want to close this
part of collisions
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with a small
discussion of units.
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And that will become
interesting and important later
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on when we look at
particle physics examples.
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So in particle
physics, we often talk
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about units of electronvolt
in collision experiments,
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or mega electronvolts,
or kilo electronvolts,
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tera electronvolts.
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So 1 electronvolt is
the kinetic energy
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of the particle with charge
e, which is accelerated
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in a potential of 1 volts.
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So that corresponds--
that's a unit of energy
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and it corresponds
to 1.6 times 10
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to the minus 19
joules or 1.6 times 10
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to the minus 90 kilograms
meter squared over 2nd square.
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But the mass of an electron
is really, really small.
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And those units
here are introduced
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because the mass
is small and you
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want to have reasonable
numbers to work with.
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So the mass of the
electron is 9.11 times
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10 to the minus 31 kilogram.
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So if you just rewrite an m0 as
equal to m0 c squared times 1
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over c squared you
find that, huh,
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now we rewrite this and find
that the masses 8 times 10
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to the minus 14
joules over c squared.
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Or in units of
electronvolts, 5 times 10
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to the 5 electronvolts
over c squared,
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which is 0.511 mega
electronvolts over c squared
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or 511 kilo electrons
over c squared.
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So when we talk about
the mass of an electron,
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we sometimes approach this
with natural units, in which c
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squared is equal to 1.
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And that just simply says
that the mass of an electron
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is 511 kilo electronvolts.
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The math of a neuron
is mega electronvolts,
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and so on, and so on.