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- [Voiceover] We've
made some good progress

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in our derivation of parts of
the Lorentz transformation.

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We've been able to
express x prime in terms

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of our Lorentz factor and x and v and t.

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And we've been able to
switch things around

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and represent x in terms
of the Lorentz factor

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and x prime and v and t prime.

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And we were able to solve
for the Lorentz factor.

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Now, the final missing
piece in order for us

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to have the full transformation
is to express t prime

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in terms of x and t.

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So, how can do we do that?

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Well, the way I'm going to
tackle it is I'm just going

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to take this equation right over here,

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let me underline it.

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I'm just gonna take that
equation right over there

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and solve for t prime.

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And the parts that have an x prime in it,

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I'm gonna substitute with this.

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So let's do that, let's
solve this for t prime.

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The first thing that I wanna do is

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I wanna divide both sides of this equation

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by the Lorentz factor or by gamma.

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So if I do that, I'm going to get,

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I'm gonna move it over to the
left so I have more space,.

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It's going to be x over gamma,

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x over gamma

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is equal to x prime.

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x prime plus v

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times t prime.

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t prime, let me do it
in that same blue color.

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Now everything I'm gonna do here

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is pretty straightforward algebra,

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but it's gonna get a little hairy,

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so that's why I wanna take some caution

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with the colors and progress slowly.

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Now, since I wanna solve for t prime,

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let me subtract x prime from both sides.

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So the left-hand side is
going to be x over gamma.

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I think that looks like a v too much.

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x over gamma minus x prime.

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Minus x prime

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is equal to v times t prime.

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v times t prime.

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And now, to solve for t prime,

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let's just divide both sides by v.

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And so, we are going to get x,

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let me do that white color.

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We're gonna get x over,

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well, now it's going to be gamma v.

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So, gamma v,

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v's in that orange color.

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Gamma v

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minus x prime

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over v

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is going to be equal to,

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is equal to t prime.

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So now we'll do what I said before.

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We've solved for t prime
in terms of now gamma v,

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x, and x prime, but now
we can take that x prime

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and replace it with gamma
and all of this business

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right over here, so let's do that.

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If we take this

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and substitute it in for x prime,

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actually, let me swap sides too,

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we are going to get t prime

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is equal to x

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over gamma.

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Let me do the gamma in that red color.

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Over gamma

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times v

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minus this stuff.

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So we're gonna have gamma times,

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it's a little bit tedious,
but we'll power through it.

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x minus vt .

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And if at any point you get inspired,

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I encourage you to run with it.

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So we just replaced x prime
with this stuff over here

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and then we're gonna
have all of that over v.

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So, all of that

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over v.

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And now what we can do, let's see.

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Let's factor out a
gamma out of everything.

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So we will get t prime

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is equal to

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gamma times,

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and we're gonna have, it's
gonna get pretty hairy now.

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Gamma times x

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over gamma squared.

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x over gamma squared, you
factor out a gamma here,

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or another way to think
about gamma times x

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over gamma squared is
gonna be x over gamma.

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We still have that v over there.

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v and then minus.

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Minus.

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And actually, let me just
distribute the minus sign,

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the negative sign.

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So, minus x over v.

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Minus x over v.

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x over v.

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Looking forward to this
getting a little bit simpler.

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And then a negative times
a negative is a positive,

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so it's gonna be plus vt divided by v.

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Well, that's just going to be plus t.

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Plus t.

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So, simplify it a little bit.

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Plus t.

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We're making some progress here.

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And so that is going to be
equal to, actually, let me just,

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so I don't have to keep
rewriting everything,

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let me just focus on this
part right over here,

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try to simplify it and
actually, even better,

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let me just focus on that
part right over there.

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That part, we can factor out an x.

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If we factor out an x,

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it is going to be equal to

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x times one over gamma squared, v.

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So let me write that down.

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One over gamma squared, v.

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Get the colors right.

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Gamma squared, v.

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Gamma squared, v,

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and then minus one over v.

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W have the minus one over v.

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One over v.

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Now, if we want to
subtract these two things,

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and let me put the close parentheses.

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If we wanna subtract these
two things, it's nice to have

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a common denominator, so
let's multiply the numerator

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and denominator here by gamma squared.

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This is gamma squared,

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gamma squared.

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And so, now I'm going to focus on

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this part right over here

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and hopefully this will simplify nicely.

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This is the same thing as

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one minus gamma squared,

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over gamma squared v.

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Over gamma, I picked a different color.

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Over gam, (chuckles) I'm having
trouble switching colors.

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Over

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gamma squared, v.

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Now what does this simplify to?

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Well, it seems like it
will be useful to have

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a different way of writing,

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well, let's just think
about what gamma squared is.

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Gamma is this business right
over here, which we could,

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if we were to square it, gamma squared.

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Gamma, that looks like a v again.

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Gamma squared is going
to be equal to one over

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one minus v squared over c squared.

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I just squared the numerator,
one squared is one.

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Take the square of the square root,

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you're just gonna get that.

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And if I wanted to
simplify it a little bit,

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I could multiply the numerator
and denominator by c squared.

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And so then that's going to be equal to

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c squared over, you
multiply the denominator

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by c squared, you're gonna get c squared

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minus v squared.

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So how does that help us?

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Well, this business.

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One, we can write as

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c squared minus v squared

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over c squared minus v squared.

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So, let's do that.

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c squared minus v squared

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over c squared minus v squared.

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I just did that so it
has the same denominator

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as gamma squared.

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And then, we're going to
subtract gamma squared.

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So we're going to subtract,

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we're going to subtract c squared

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over c squared minus v squared.

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And it looks like this is
going to simplify nicely.

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Well, let me just do one step at a time.

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And we're gonna have all of that over,

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gamma squared

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is once again c squared

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over c squared minus v squared.

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And we're gonna multiply that times v.

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Times, the v in that same color,

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all of that times v.

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Now, let's see, up here in the numerator,

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I have c squared minus v
squared minus c squared.

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So, what's going to happen is

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this, and we have the same denominator,

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so this and that are going to cancel.

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And so this whole expression
is going to simplify to

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negative v, and let me do
it in that same v color.

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Negative v squared

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over c squared minus v squared.

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c squared minus v squared.

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And we're dividing by this

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and that's the same thing as
multiplying by the reciprocal,

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so let's multiply b the reciprocal.

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So, times c squared minus v squared

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over c squared times v.

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And we are now in the home stretch.

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c squared times v.

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And we could do some simplification now.

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That's going to cancel with that.

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And then, v squared divided by v,

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you're just gonna be left
with a v right over there.

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So, all of this crazy business

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has simplified to a negative v,

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and I'll just write it
in that orange color.

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It has all simplified to negative v

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over c squared.

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So all of this

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has simplified to
negative v over c squared

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and so we're in the home stretch.

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This expression, t prime,

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it's going to be equal to gamma times,

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let's write this t first.

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So, t .

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And then this expression has simplified

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to negative v over c squared times x.

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So we can write this as

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minus v over c squared

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times x.

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And we're done.

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We have just completed our
Lorentz transformation.

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We started with this, that
we've been able to show

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in the last few videos, and
we did a little bit of hairy,

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carefully, did a little
bit of hairy algebra

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to get this result: t
prime is equal to gamma

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times t minus v over c squared times x.