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We would like to now introduce
a new methodological tool
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for analyzing problems that
involve momentum transfers.
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And we call that tool
momentum diagrams.
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Now, what we'd
like to do is look
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at our fundamental idea, which
was, for discrete objects,
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we have that-- involved
in a collision-- we have
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that the external force
integrated with respect
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to time-- in poles-- is equal
to the change in momentum
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between two different states.
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So this is the momentum
of the final state,
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and this is the momentum
of the initial state.
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So when we want to
analyze problems,
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how can we methodologically
introduce a picture
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representation of our problems?
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So let's look at what
we need to do first.
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We always need to choose a
system that we're referring to.
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And with respect to the
choice of that system,
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we need to also choose
a reference frame.
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Now, once we've done
that, we can now
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represent a collision with
respect to this system.
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For instance, let's consider
two objects-- as our system
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Object 1 and Object 2.
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They're moving on a
frictionless, horizontal
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surface.
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And we're choosing as a
reference frame, the ground
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frame.
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Now, what we'd like to do
is identify two states.
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So for our initial
state, we'll have--
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if we're given some
initial conditions-- what
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we'd like to do is represent
each object with a velocity
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vector.
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So here, we'll
write the 1 initial,
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and let's suppose this object
is coming at it with the 2
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initial as an example.
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And once we've represented
the velocities of the objects,
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we can write down their
momentum in the initial state.
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Similarly, if these two objects
collide and they're moving,
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we actually don't know which
way those objects will end up
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moving.
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And so what we'd again like
to do in our final state,
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after this collision, is to
represent the velocities by,
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again, vectors.
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So this is V1 final
and this is V2 final.
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And then, we can represent
the change in momentum.
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So our momentum principle
now becomes-- in this case,
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let's just assume that
the external force here
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sums to zero-- we're
assuming no friction.
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And then our momentum
principle says that 0
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equals P final minus P initial.
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And now, we can read
off those momentums
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as vectors on the diagram,
and so what we have
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is that the final
momentum will be
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equal to the initial momentum.
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So we can write down
M1 final plus M2
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V2 final is equal to M1 V1
initial plus M2 V2 initial.
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And that's how we can represent
a collision where there
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is no external forces and
use our momentum principle
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to get an vector equation.
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Now, in many problems,
you're given information.
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You might be given information
about the speeds and magnitudes
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of the objects.
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And in order to then
take this equation
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and represent it in speeds and
directions or even components,
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we need to choose some
coordinate system.
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So if we choose a coordinate
system-- and that's
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the third step-- so suppose
we choose a coordinate system,
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then we can start to look at
two different representations
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for our problem.
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For instance, let's just choose
this to be the i hat direction.
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Now, given that choice, we
could describe the velocities
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in terms of components.
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Now this gets awkward--
V1i x component i hat.
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And similarly, we can write
down all the velocities-- V2i--
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as V2ix i hat, et cetera,
for all the velocities.
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And our momentum
equation in components
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then becomes, in the i
hat direction, M1 V1ix
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plus-- well, here we
have the final state,
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so let's make this consistent--
the final plus M2 V2 final x
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equals M1 V1 initial x
plus M2 V2 initial x.
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And that's the same equation
that we have as vectors, now
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expressed in terms
of components.
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And recall that components
can be positive or negative.