00:01
Okay, so problem number 62 wants us to figure out the kinetic energy that a proton has to have in order to be able to neglect or to not include its rest mass when calculating its wavelength, if we want its wavelength to be within 1%.
00:19
So the first thing we need to do is calculate the actual wavelength of it.
00:26
And in order to do that, we need to know the energy, the total energy of the protons.
00:32
So let's go ahead and start with the fact that so the total energy is equal to the kinetic energy plus the rest energy squared.
00:48
And the relativistic energy term is given by the momentum times the speed of light squared plus the rest mass squared c to the fourth to the rest mass energy squared.
01:01
Now we can rearrange this and put it in the term p squared c squared is equal to the total energy minus the rest mass energy to the fourth.
01:14
And now we have this for the total energy here.
01:22
So we can plug that in here, and we get p squared, c squared, is equal to k plus m .p.
01:36
Mast the proton times c squared, squared, minus mp squared c to the fourth.
01:44
So now expanding this, you get p squared, c squared, is equal to k squared, plus 2k mpc squared minus plus mp squared c to the fourth minus mp squared c to the fourth right those cancel so now we have p squared c squared is equal to k squared plus 2k 2k mpc squared all right so now we can let's go ahead and pull out the kinetic energy to simplify this, so which is equal to, we get k squared times 1 plus 2mpc squared over k is equal to p squared so if we take the square root of both sides, we get p c is equal to k times the square root 1 plus 2 mass of the proton c squared over the kinetic energy.
03:03
All right.
03:04
So the reason we've done all of this is now if we look at the debroli wavelength, so we want to calculate the wavelength.
03:11
So the actual wavelength, we're going to denote lambda 0, which is the true wavelength.
03:17
So that's equal to h over p.
03:19
So now we have p is equal to so we have p times c is equal to this term so if we divide both sides by c we end up with and plug in the momentum we end up with h the wavelength is equal to hc over k times one plus mass proton c squared over k or, oh, and i should say, since we want to know at what kinetic energy, the, it has to be at, the proton has to have in order to be within 1%.
04:10
And say lambda, which is the calculated, the one where we ignore, the rest of mass energy is equal to h, c, okay, where k is the kinetic energy, but since we're ignoring, it would actually be hc over e, because you have e is equal to h.
04:39
E is equal to hc over lambda.
04:43
So lambda is equal to hc over e.
04:50
But we're only wanting to, so that would be the total energy, but we're wanting to ignore the mass energy.
04:56
So we're just saying only the kinetic energy.
04:59
So that's going to be greater than lambda zero, because it's going to have less contribution.
05:09
So this number is going to be smaller, so this number is going to be bigger.
05:14
And then, lambda, we want it to be within 1 .01, so 1 % lambda 0.
05:24
So if we look, so that means it's going to be 1 % larger within 1 % of lambda 0.
05:32
So now we can plug all of this in and say lambda is equal to hc over k, it's going to be equal to 1 .01 times lambda not, hc over k times 1 plus mpc squared over k to the one -half.
05:57
The k is cancel, the hc is cancel.
06:02
We end up with 1 over 1 .01 is equal to, 1 over the square root of 1 plus mpc squared over k so 1 .01 is equal to square root of 1 plus mpc squared over k.
06:31
So we want to solve for k so now we'll square both sides 1 .01 squared squared and then rearrange it a little bit.
06:48
So we'll move, i'll just do this, is equal to 1 plus, i don't want to skip too many steps for you guys, mp squared over k.
06:58
So now, just move the 1 over that way, so 1 .01 squared minus 1 .1, it's equal to mpc squared over k.
07:11
Solve for k multiply that side by k divide that side up here so we get k is equal to you know i just realized i've been i haven't carried a two so there's a two up here a two there and i lost it so again when you're doing this kind of math you got to really make sure that you don't miss something like that it's supposed to be a 2.
07:47
Because yeah, you make a mistake somewhere where you forget a number and it just carries you, you just end up going along with it and you end up with the wrong answer at the very end.
07:55
So you have to be really careful about that kind of thing.
07:57
So now, so k is equal to 2 mpc squared divided by 1 .01 squared minus 1.
08:10
Which now we can plug in the numbers that we have, which is equal to 2 times 9 .38 times 10 to 8 electron volts, or 938 mega electron volts, divided by 0 .0201, which gives us an energy, i've got a room here, 9 .333 times 10 to the 10th electron volts for 933.
08:56
Sorry, not 933.
09:05
93.
09:05
93.
09:08
So it needs to have kinetic energy of 933 .3.
09:14
So it needs to have kinetic energy of 933 .3.
09:17
Giga electron volts in order to be able to not include its rest mass and still be within 1 % of and still be able to calculate its wavelength within 1%.
09:33
So now there's a couple more parts of this problem.
09:36
It wants us to calculate, it wants us to calculate the corresponding wavelength and the speed of the proton.
09:45
So i'm going to create a new page here.
09:49
So now it's calculated wavelength...