WEBVTT
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We would now like to look at
two-dimensional collisions.
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And what we'd like to look at
is in the laboratory frame--
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so I'll call that the lab
frame-- in which we have
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a target particle, which
I'm going to call 2,
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and an incoming particle
1, which is coming in
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with some initial velocity.
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After the collision,
let's imagine
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that the target
particle is going out
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at a certain direction.
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So we'll call that 2.
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And the target particle
has a velocity v2 final.
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And the initial particle that's
going in this direction-- we'll
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call that 1.
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And that is it's
outcoming velocity.
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Now in this collision, we
want to ask ourselves first,
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what quantities are
constants of the motion.
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Well, let's assume
no external forces,
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therefore momentum is constant.
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And we can write our
momentum equation
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as m1 v1 initial equals m1
v1 final plus m2 v2 final.
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Now recall that
momentum is a vector.
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And so what we have here
are two-- the unknowns here
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are our two outcoming vectors,
v1 final and v2 final.
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And a vector in two
dimensions has two quantities.
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We can discuss-- we can
write that as components.
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Or we can write it in terms
of magnitudes and directions.
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Now experimentally, we often
will measure the directions
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of the outcoming
particles, which
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I will now indicate by theta
2 final and theta 1 final.
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And so, when we write our
two momentum equations,
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we can either write
it as components,
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or we can write it in terms
of magnitudes and directions
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and do vector decomposition.
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Now because we measure
the outcoming directions,
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we're going to choose to do
magnitudes and directions.
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So let's indicate
a little notation.
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We'll say that the magnitude
of v1 initial is v1 i.
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And the magnitude of
v2 final is v2 final.
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And the magnitude of
v1 final is v1 final.
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And so now, when we look at
our two momentum conditions,
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we can-- we now also
have to introduce
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unit vectors for directions.
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So let's call i hat that way
and j hat in this direction.
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And in our i hat
direction, we have
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only the incoming momentum.
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And we can write that
as m1 v1 initial.
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It's positive because we've
chosen the forward direction
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as our i hat direction.
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Now in terms of the outgoing
momentum in the i hat
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direction, we have to
do vector decomposition
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of both of these vectors.
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And they both have
positive components.
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So we have m1 v1 final-- that's
the magnitude-- cosine theta
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1 final plus-- positive
sign, because they're
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both in the positive
direction-- v2 final magnitude
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times-- we need that little-- m2
v2 final cosine theta 2 final.
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And that is our i hat direction.
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Now the j hat
direction-- remember,
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we have to be careful, because
we're taking positive j hat up.
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So our particle 2 has a positive
component in the j direction.
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And our particle 1 has
a negative component
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in the j direction.
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The incoming momentum-- there's
no momentum in the j direction.
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So we have a 0.
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And that's equal to
positive m2 v2 final.
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And that's a sine theta 2 final.
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Now, here's where you
have to be careful,
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because this one is negative.
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Component is in the
negative j hat direction.
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And we have m1 v1 final
sine theta 1 final.
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And these two represent
our momentum equations.
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Now we also have to think--
let's think about energy.
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Again, we have to know
something about this collision.
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And our assumption will be
that this particular collision
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is elastic.
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And that means the
initial kinetic energy
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is equal to the
final kinetic energy.
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Energy is a scalar.
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We've been describing our
incoming velocity vectors
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in terms of magnitudes.
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So we can write our
elastic energy condition
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as the incoming kinetic
energy squared--
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that's the kinetic
energy incoming--
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is equal to 1/2 m1 v1
final squared plus 1/2
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m2 v2 final squared.
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And that is our kinetic
energy condition.
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Let's label this equation
1 and equation 2.
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Now, it's very
important to realize
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which quantities are given and
which we need to solve for.
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So in this problem, because
the two outcoming velocities--
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unknowns-- we have
four unknowns.
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Those unknowns can be written
in terms of the two velocity
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final and the other
one, v2 final.
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Those are our vector quantities.
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But recall, in terms of
the scalar magnitudes,
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we have that v1 final.
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And I'll just write the
other ones down-- v2 final,
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and the two outgoing
directions-- theta 1 final
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and theta 2 final.
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So these are our four
unknown quantities.
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But you can see we only
have three equations.
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And therefore, if we want
to determine the outcomes,
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we need to measure one
additional quantity.
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Now that's very useful
when doing problem solving,
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because when you start
to read a problem,
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and you look at
what's being measured,
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you can right away determine
which of the four quantities
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has been given.
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You may be given an outgoing
magnitude of the velocity,
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or you may be given one of
these scattering angles.
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And so that's how we approach
two-dimensional elastic
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collision.
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Of course there's
algebra now to solve
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for any particular quantity
that you're interested
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in, provided you have this
extra additional information.