So, how does this play into things that we can measure like momentum? Well we remember what momentum was, according to Newton's model, which we can call classical, P is just M times V. Okay, if you double V, you double the amount of momentum, but you can't just keep doubling V forever and ever because there's this universal speed limit to the world that we live in, which is C. All right, you can't take V up above C, something bad happens when you do that. And so we need the relativistic version of momentum, which looks almost the same except we're gonna stick a gamma right in front of it. Alright, and again, gamma is equal to 1 over the square root, 1 minus V squared over C squared. Okay, it's a number bigger than 1. So relativistic momentum has more momentum in it than classical momentum. All right, so let's think about this now in terms of one of your homework problems and this is -- let's see if I can figure out what number it is. Okay, this is one that we looked at earlier in office hours, it is 26.18 and it says the following. Let particle of mass M move at some V, we'll call it V1 of 0.37 C. How fast does V2 have to be to have twice the momentum? Okay, so particle moving at V1, it's got some momentum, we want to go faster to V2 and we want to hit twice the momentum. So, if you were in the classical world obviously you would just double it, right? Double V, it doubles the momentum. But now, in this relativistic world we have this factor gamma which has V in it. All right, so we have two relationships, we have P1 is equal to gamma 1 mV1, and then we have P2, which is equal to gamma 2mV2, and we know that that is going to be twice the initial momentum P1. All right, so now we have this relationship here and we need to solve for V2. All right, so let's see if we can do that. Alright, so let's write it over here we've got two times P1 is equal to gamma 2mV2, and we know what P1 is it's gamma 1mV1. Alright, right off the bat we can get rid of m. Okay, m is the rest mass, they were calling it M naught, it's the same thing, M is the rest mass. Those cancel out -- those cancel out and now let's do a little trick, let's divide by gamma 2 and let's divide by gamma 1 on the right side, okay? So, I took gamma 1 and put it under there, I took gamma 2 and put it under there, and now let's square both sides. So if I square this and I square this, what do we get? We get 4 V1 squared over gamma 2 squared, equals V2 squared over gamma 1 squared. Okay, but we know that gamma 2 squared is 1 over the square root of 1 minus V squared over C squared, so 1 over gamma 2 squared has to be 1 minus V2 squared over C squared. 1 over gamma 1 squared is going to be 1 minus V1 squared over C squared. And so now this whole thing here becomes the following, we get 4 V1 squared times 1 over gamma 2 squared, which is 1 minus V2 squared over C squared, and on the right side we have V2 squared times 1 over gamma 1 squared, which is 1 minus V1 squared over C squared. And now we can just simplify this and solve for V2. All right, let's just do it real quick. Multiply everything by C squared we get 4 V1 squared times C squared minus 4 V1 squared V2 squared, equals V2 squared C squared minus V2 squared V1 squared, and now I have a V1 squared, a 2 V2 squared -- I have a V1 squared, a 2 V2 squared and so I can put those together, and then I have 4 V1 squared C squared that's left. When I add 4 to that negative 1, I get 3 V1 squared V2 squared and now we can put the V2's together, V2 squared times C squared plus 3 V1 squared and then finally we can divide by that quantity and you end up with the following. V2 is equal to -- and I'm going to take the square root, so I can do that right now, 2 V1 C, all over the square root of C squared plus 3 V1 squared. Okay, I went through that slightly quickly but you can replay the video later on and double-check your algebra and make sure you get the same thing, and now you have all these numbers, right? We have V1 we have C obviously and if you plug in all those numbers, you can double check with mine, you should get 0.62, and it wants it in the units of speed of light, and so you write 0.62 and then the units are C. Okay, so those were for my numbers, you of course will have different numbers for your V1. That one look familiar Jessica? Yeah. That's the one we went over earlier in office hours, painfully, I'm a add. All right, questions about this one before -- before we move on?