36. Special Relativity

Consequences of Relativity

Physics36. Special RelativityConsequences of Relativity

Hi guys, welcome back. Let's take a look at one of your homework problems that relates to the energy problem that we just talked about, and this is number 2629 and it says how much work is required to accelerate a proton from rest, up to a speed of 0.984C. Okay, so they give us the following, the initial speed is zero, the final speed is 0.984C, and they want to know how much work is required to do that. So let's think about a proton, here's a proton sitting there and now we're gonna start pushing on it and eventually it's going to be moving along at speed V final. When it is sitting there initially, how much energy is in the system? Well, it's not moving so there's no momentum there but there is of course the rest energy of the proton. When it starts moving it now has a final energy which is gamma MC squared. Okay, so this gamma thing looks really critical to understanding what the total energy is, and if we can calculate gamma, we can probably figure out how much work is required because the work is just going to be the change in energy. We know what the final is, its gamma MC squared. We know what the initial was, it's just MC squared, and so we get gamma minus 1 times MC squared. So let's figure out what gamma is. Gamma is 1 over the square root 1 minus V squared over C squared, and if we are moving at 0.984 C, what do we get for our gamma? We have a 0.984 C, we're going to divide by C and square that stuff, therefore the C's will cancel out, we get 1 over the square root 1 minus of 0.984 squared, and why don't you guys punch that into your calculator and tell me what you get. Anybody get an answer for that one? >> (student speaking) 5.6. >> 5.6. Okay, somebody else confirm that, 5.6? Okay, so work we said was gamma minus 1 MC squared, and so it's 4.6 times M times C squared, but what is M? M is the mass of a proton. It's the rest mass of a proton and if we flip open the inside cover of your book you can figure out what the rest mass of a proton is, it's 1.67 times 10 to the minus 27 kilograms. We're going to multiply by C squared three times 10 to the 8th, we're all in SI units here and so we can punch this into our calculator and see what we get. >> (student speaking) I have a question. >> Yeah? >> (student speaking) Where did you get the 4.6? >> 5.6 minus one, yeah. And we got to square the C, right? Yeah. And if somebody gets a number, shout it out. >> (student speaking) 6.9 times 10 to the minus 10. >> 6.9 times ten to the minus ten and we're in SI units so that should be joules, and that's what I get here on Wolfram Alpha so let's try that, let's punch it in and see if that's what we get. 6.9 times ten to the minus ten -- good, sounds correct. Okay, that doesn't sound like a lot of energy but remember a proton is really really pretty small, right? It's ten to the minus 27 kilograms, so that's a pretty significant amount of energy for moving that thing, that fast. Okay, Part B says, what would be the momentum of this proton? All right, that's not too bad. All right, momentum is what? Well, momentum is gamma mV. Okay, so in this case it's going to be gamma MV final and for our numbers we already know what gamma is. Gamma was 5.6, mass of the proton 1.67 times ten to the minus 27 kilograms, V final is right there, 0.984 C, which is three times 10 to the eighth. All right, punch all those numbers into your calculator and let's see what we get. And I got 2.76 times 10 to the minus 18. Anybody else get that one? Okay. What are the units of momentum, remember its kilogram meters per second, right? Gamma is unit-less, mass is kilograms, V is of course meters per second. So let's try that one and see if we're right. And I think it needs a dot in there and that says we're right, okay? So try that with your numbers, you will of course have a different value for your V-final, which means you will have a different value for your gamma. Questions on that one? Everybody okay? Let's ask you a follow-up question, how much momentum would it have classically? That one's easy, right? Classically P is equal to mV and so it's the mass of the proton, 1.67 times ten to the minus 27 times this V, 0.984 times C, three times ten to the eighth. That's going to be a number that is significantly less than the one that we just calculated. Right, and you know exactly how much less, if I just take away the gamma and do that calculation, what do I get? I get what I had before, 2.76 times ten to the minus 18 divided by the gamma that we calculated for ours, which was 5.6. Okay, so relativistic momentum takes significantly more oomph to get these things going, significantly more energy to get these particles going and it's because of that gamma factor that it keeps increasing.