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- [Voiceover] Now let's apply the formula

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we came up with the last video

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sometimes known as the Einstein
velocity addition formula.

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And we'll see that it's
a pretty neat thing.

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So let's say, so this is once
again me floating in space,

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my frame of reference is just
the s frame of reference,

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Let's say my friend over here,

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she's flying away from me, her velocity,

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let's give it some numbers.

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Let's say she's flying away from me

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with a velocity of, let's say,

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0.7c, so seven tenths
of the speed of light

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or 70% of the speed of light.

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And let's say this third
character right over here,

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let's say they're flying towards me,

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so let me change the
direction, the velocity vector.

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So let me erase that.

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Gee, that doesn't look like a pure black

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but it looks fairly
erased, so there you go.

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And let me give the velocity
vector in the other direction.

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So the velocity vector,

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this character is flying towards me

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at let's say, so u is
going to be equal to,

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so it's flying towards me

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x is decreasing with time,

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let's say it's negative,

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- 0.5c,

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so flying towards me at
half the speed of light.

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And once again, both of these velocities

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are given in my frame of reference.

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Now if we were in a Newtonian world

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or if we used the
Galilean transformations,

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and this is released as
something on your highway.

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If I'm on a highway and if
I'm going in one direction

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at 70 miles an hour, and
someone is coming towards me

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at 50 miles an hour, it'll look to me

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like they are coming towards
me at 120 miles per hour.

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We would essentially
add these two together.

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And so, if we were in a Newtonian world,

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this person is flying at 0.7c

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and this person is
flying this way at 0.5c,

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you would, in a Newtonian world, say okay

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from this point of view it will look like

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this blue character is approaching

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at 0.7c plus 0.5c

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or at 1.2 times the speed of light

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which we know violates
the laws of the universe.

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And that's where this
formula is really handy.

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Actually let me even
make the spaceship point

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in this direction just
so we don't get confused.

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So the spaceship is going

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in that direction just like that.

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And once again, the
magnitude of the velocity

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coming towards me is
half the speed of light.

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I put the negative there to show

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that it's coming towards me,

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that x is decreasing over time.

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Well, lucky for us in the
last video, we came up with

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this Einstein velocity addition
formula, so let's apply it.

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From this friend's frame of reference,

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the s-prime frame of reference,

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the velocity of this character

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which is change in
x-prime over change in t

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is going to be this.

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So what do we have here?

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So u is the blue velocity
in my frame of reference,

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So that is -0.5c.

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v is my friend's frame of reference,

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the frame of reference

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that we're trying to get the velocity in.

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So that's 0.7c,

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so let me do it in that color.

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So that is 0.7c.

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And then, on top of uv over c-squared,

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so once again, u is

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- 0.5c.

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I know I wrote it really small.

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And then, v is

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0.7c.

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So what is this going to be equal to?

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Well, the numerator -0.5c

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minus 0.7c,

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that's going to be negative.

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Our numerator, right over here

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is going to be

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- 1.2c.

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And this is the velocity
that you would expect

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if we were dealing in a Newtonian world.

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This person says, "Hey, this person looks

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"like they're coming towards me

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"at 1.2 times the speed of light."

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That's what our Galilean
transformations would give us.

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But luckily, we have all
this business at the bottom

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that keeps us from
violating the absoluteness

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of the velocity of light,

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this notion that nothing can travel faster

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than the speed of light.

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Because in our denominator,

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we're going to get 1 minus 0.5c

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times 0.7c.

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Let's see, 0.5 times 0.7

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is going to be 0.3.

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So 0.3, I could say 0.30 if I liked,

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but I'll just write 0.3c squared.

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It'll be a negative.

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I have a negative times a positive.

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So I could put the negative there

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but if I'm subtracting a negative,

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that's just going to be a positive.

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And I'm just going to divide by c-squared.

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And so, that cancels out.

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And so notice, we're going to be dividing

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- 1.2c by something that's
slightly larger than 1.2.

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So this is going to be

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- 1.2c

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over 1.3.

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1.3

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And so, lucky for us, this
is going to be less than c,

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or this absolute value is
going to be less than c.

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So it's going to be, let's
get a calculator out.

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So, if we have

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1.2 divided by 1.3,

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this is equal to approximately 0.92.

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So this is going to be approximately

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- 0.92c

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which is cool.

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It kind of goes with our,

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let me write it over here.

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So this delta x prime over delta t prime

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is -0.92c.

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So it makes sense that from our
friend's frame of reference,

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this ship will look like
it's approaching her

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faster than from my frame of reference.

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It looks like it's approaching
me in my frame of reference,

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it looks like it's coming to
me at half the speed of light.

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In her frame of reference,

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it looks like it's coming at
0.92 times the speed of light.

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The negative is just
specifying the direction.

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But we didn't violate
the fundamental postulate

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of special relativity,
that nothing can go faster

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than the speed of light.

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The speed of light is absolute.

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So this is pretty cool.

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I've often even thought about
making a video game somehow

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where you leverage this,

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where things are flying in
different relative velocities

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but special relativity applies.