Applying Einstein velocity addition

0:00- [Voiceover] Now let's apply the formula we came up with the last video sometimes known as the Einstein velocity addition formula. And we'll see that it's a pretty neat thing. So let's say, so this is once again me floating in space, my frame of reference is just the s frame of reference, Let's say my friend over here, she's flying away from me, her velocity, let's give it some numbers.

0:19Let's say she's flying away from me with a velocity of, let's say, 0.7c, so seven tenths of the speed of light or 70% of the speed of light. And let's say this third character right over here, let's say they're flying towards me, so let me change the direction, the velocity vector.

0:38So let me erase that. Gee, that doesn't look like a pure black but it looks fairly erased, so there you go. And let me give the velocity vector in the other direction. So the velocity vector, this character is flying towards me at let's say, so u is going to be equal to, so it's flying towards me x is decreasing with time, let's say it's negative,

1:03- 0.5c, so flying towards me at half the speed of light. And once again, both of these velocities are given in my frame of reference. Now if we were in a Newtonian world or if we used the Galilean transformations, and this is released as something on your highway.

1:21If I'm on a highway and if I'm going in one direction at 70 miles an hour, and someone is coming towards me at 50 miles an hour, it'll look to me like they are coming towards me at 120 miles per hour. We would essentially add these two together. And so, if we were in a Newtonian world, this person is flying at 0.7c and this person is flying this way at 0.5c, you would, in a Newtonian world, say okay from this point of view it will look like this blue character is approaching at 0.7c plus 0.5c or at 1.2 times the speed of light which we know violates the laws of the universe.

1:59And that's where this formula is really handy. Actually let me even make the spaceship point in this direction just so we don't get confused. So the spaceship is going in that direction just like that. And once again, the magnitude of the velocity coming towards me is half the speed of light.

2:15I put the negative there to show that it's coming towards me, that x is decreasing over time. Well, lucky for us in the last video, we came up with this Einstein velocity addition formula, so let's apply it. From this friend's frame of reference, the s-prime frame of reference, the velocity of this character which is change in x-prime over change in t is going to be this.

2:38So what do we have here? So u is the blue velocity in my frame of reference, So that is -0.5c. v is my friend's frame of reference, the frame of reference that we're trying to get the velocity in. So that's 0.7c, so let me do it in that color. So that is 0.7c.

3:05And then, on top of uv over c-squared, so once again, u is

3:11- 0.5c. I know I wrote it really small. And then, v is 0.7c. So what is this going to be equal to? Well, the numerator -0.5c minus 0.7c, that's going to be negative. Our numerator, right over here is going to be

3:36- 1.2c. And this is the velocity that you would expect if we were dealing in a Newtonian world. This person says, "Hey, this person looks "like they're coming towards me "at 1.2 times the speed of light." That's what our Galilean transformations would give us.

3:53But luckily, we have all this business at the bottom that keeps us from violating the absoluteness of the velocity of light, this notion that nothing can travel faster than the speed of light. Because in our denominator, we're going to get 1 minus 0.5c times 0.7c.

4:15Let's see, 0.5 times 0.7 is going to be 0.3. So 0.3, I could say 0.30 if I liked, but I'll just write 0.3c squared. It'll be a negative. I have a negative times a positive. So I could put the negative there but if I'm subtracting a negative, that's just going to be a positive.

4:36And I'm just going to divide by c-squared. And so, that cancels out. And so notice, we're going to be dividing

4:42- 1.2c by something that's slightly larger than 1.2. So this is going to be

4:49- 1.2c over 1.3. 1.3 And so, lucky for us, this is going to be less than c, or this absolute value is going to be less than c. So it's going to be, let's get a calculator out. So, if we have 1.2 divided by 1.3, this is equal to approximately 0.92. So this is going to be approximately

5:23- 0.92c which is cool. It kind of goes with our, let me write it over here. So this delta x prime over delta t prime is -0.92c. So it makes sense that from our friend's frame of reference, this ship will look like it's approaching her faster than from my frame of reference.

5:49It looks like it's approaching me in my frame of reference, it looks like it's coming to me at half the speed of light. In her frame of reference, it looks like it's coming at 0.92 times the speed of light. The negative is just specifying the direction. But we didn't violate the fundamental postulate of special relativity, that nothing can go faster than the speed of light.

6:07The speed of light is absolute. So this is pretty cool. I've often even thought about making a video game somehow where you leverage this, where things are flying in different relative velocities but special relativity applies.