WEBVTT
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This is College Physics Answers
with Shaun Dychko.
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We are going to choose some relative
velocity between two observers
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and one of the observers
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emits some light away from
the other observer
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which of course has a speed of *c*,
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according to this particular observer.
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And the speed of this light measured by
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this moving observer because we have chosen
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to have this observer on the right, at rest.
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And we'll say that the observer on the left
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is the reference frame that is moving
with some velocity *v*.
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And the speed that this person
that is moving
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observes for the light is going
to be labeled *u prime*.
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And so we have this relativistic
addition of velocities formula
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which says that the velocity
observed by the
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person at rest is going to be
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the velocity with which the
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other observer is moving plus the
velocity that they report
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divided by 1 plus *v u prime*
over *c* squared.
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So we are taking the right to be positive
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and that means this velocity—*u prime*—
is going to be negative *c*
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because it's towards the left.
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And let's see what *u* observes.
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So we plug in negative *c*
in place of *u prime*.
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So *u* is *v* plus negative *c*
over 1 plus v
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times negative *c* over *c* squared.
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So on the top it's *v* minus *c*
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and on the bottom, we have
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one of the *c*'s in the bottom
canceling with this
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*c* at the top and that makes
negative *v* over *c*.
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Then we multiply top and bottom by
*c* over *c*
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and on the bottom, we have
1 times *c* is *c*.
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And then minus *v* over *c*
is just going to be *v*.
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Then on the top, we have
*c* times *v* minus *c*.
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Then let's multiply this by negative 1
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and then multiply the *c* by negative 1.
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So we are multiplying by 1, in other words,
negative 1 times negative 1
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and this bracket when it's multiplied by
negative 1 ends up
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having the signs changed inside
so it's *c* minus *v*
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and this negative 1 stays there
so negative *c*.
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And so we have created binomials that
are the same and these cancel.
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And so we have the *u* is negative *c*.
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And so quod erat demonstrandum,
we have shown
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that the velocity of this light
measured by this person
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is gonna be negative *c*,
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just the same as the velocity measured
by the moving observer.