WEBVTT
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So we analyzed our
one-dimensional collision
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where we had an object 1 moving
with some initial velocity,
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object 2 also moving with
some initial velocity.
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And after the collision,
we just arbitrarily
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said object 1 is moving
this way and object 2
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was moving that way.
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And we called that
our i hat direction.
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In this collision,
we assume that it
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was a frictionless surface.
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And there were no
external forces.
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So momentum is constant.
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Now what about energy?
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Well, we know that if there
is no external forces, then
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in principle there could
be no external work.
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However, during the
collision, we've
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established different
types of collisions.
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We said elastic
collision is when
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we are assuming that the kinetic
energy of the system-- that's
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K system final minus
K system initial--
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is 0, the statement that the
kinetic energy is constant.
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We have an inelastic
collision in which
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delta K system has decreased.
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And that can come from some
deformation of the objects
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during the collision.
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Heat is generated.
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Some sound, some
other source of energy
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in which energy is
always constant,
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but the kinetic energy of
the system could diminish.
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And we also talked about
a super elastic collision.
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And in this type of
collision, the kinetic energy
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of the system has increased.
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Now that's a little bit
tricky in terms of what
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we're calling our system.
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But imagine that when
these two objects collided,
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there was some type of
chemicals on it that exploded.
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And so that was
some chemical energy
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that was converted into
extra kinetic energy.
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So in order to make a
model of our problem,
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we have to beforehand make some
assumptions about the nature
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of the collision.
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Now for our
particular collision,
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let's assume that the
collision is elastic.
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So we have two
conservation principles.
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We have that the kinetic energy
of the system is constant.
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And we've already said
that because there's
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no external forces, the momentum
of the system is constant.
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So now we can write down our
kinetic energy condition.
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We have m1 V1-- now here, this
can be a little bit tricky
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how we're going to write
this, because remember
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kinetic energy is
a magnitude squared
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plus 1/2 m2 V2 initial squared
equals 1/2 m1 V1 final squared
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plus 1/2 m2 V2 final squared.
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Now because a
component, although it
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can be positive or
negative, in one dimension,
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if you square the component,
you get the magnitude.
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So I can also write
this equation--
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and I'm going to divide
through by the halves,
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cancel all those-- as 1x initial
squared equal to the magnitude
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squared plus m2 V2x
initial squared equals M1
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V1x final squared plus
1/2 m2 V2x final squared.
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And we've already
canceled all the halves,
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so we have no half there.
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Now given this equation--
we'll call this equation
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1-- we can also had
our momentum equation,
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which I'm going to
write down as 2, m1 Vx
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initial plus m2 V2x initial
equals m1 V1x final plus m2 V2x
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final, reminding
us of our condition
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of the constancy of momentum.
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So now, if you look at this,
if we're given the masses, all
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the mi's and we're also
given the initial state
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of the system, V2x
initial, then we
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have two equations
and two unknowns.
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And we can solve algebraically
for V1x final and V2x final.
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And by determining the
signs of these components,
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we can figure out the actual
final state of the system.
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So this is going to end up
involving a quadratic equation.
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And what we'd like to do now
is show an alternative way
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that gives us another kind of
conservation, another principle
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to analyze these one
dimensional collisions.
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So we'll begin that.