00:01
In this problem, we have a particle with energy 5gv, momentum 3gv per c.
00:07
We're going to be looking for the mass, first off.
00:10
Units with these types of units, the mass will be gev per c squared.
00:15
You're going to see that the c does not participate in the calculation of any numbers, but in the manipulation appears with the units, as you'll see.
00:28
So how do we get the mass? well, there's a relativistic relationship between energy, momentum, and mass.
00:34
E squared is equal to p squared, c squared, plus m squared, c to the fourth.
00:43
So let's do some manipulations, m squared, c to the fourth, is equal to e squared, minus p squared, c squared.
00:52
And now solving for m.
00:53
Now most of my algebraic work is really going to be involving getting the c in the right spot so we can see the units clearly.
01:03
So m is equal to, 2 square root e squared minus p squared c squared over c to the fourth 1 half power e squared over c to the 4th minus p squared over c squared 1 half and now we can rewrite this e over c squared squared so notice now you might say why you're doing it g e v per c squared i told you these c's are going to go with the units so this is g ev per c squared that's what we want because that's going to be squared then we have a square root but we'll see that in a second all that then minus p over c squared all to the one -halfs now we've set everything up properly so now we put in the units like i said five is it going to be five gev per c squared, squared, minus three gev per c squared squared.
02:26
Take the one -half.
02:30
So this is 25 minus 9, 16.
02:33
Square over to 16 is 4.
02:35
4, and notice our units, g .e .v.
02:38
Squared over c to the fourth.
02:40
So take the square root, it's gev over c squared.
02:46
There you have it.
02:47
That's what i told you the units of mass would be.
02:50
So we're not putting in three times 10 to 8 for the m, the c to get mass.
02:56
You'd need that if you were getting mass in terms of kilograms or something.
03:00
Then you have to be converting the gev also.
03:03
But we're not worrying about that here.
03:08
B, it wants to know the speed this particle is traveling.
03:12
Now, the total energy is the rest of energy divided by the square root 1 minus v squared over c squared.
03:23
So some algebra, multiply both sides by the square root, divide both sides by e.
03:28
I get square root.
03:29
1 minus v squared or c squared is equal to mc squared over e.
03:38
Square of both sides, 1 minus v squared over c squared is equal to mc squared over e squared.
03:50
Solving for v squared over c squared is equal to 1 minus mc squared over e squared.
04:03
V over c is equal to 1 minus mc squared over e, square that and take the square root.
04:20
So now we can solve for v, and i'm going to put the c on at the end.
04:28
One minus.
04:30
Now let's look at the units here.
04:32
Gv, this is like a density, gv per c squared, but i've got to multiply by c squared.
04:36
It's like having kilograms per cubic meter.
04:42
Multiply by a certain number of cubic meters, you got the mass.
04:46
Well, i multiply by a certain number of c squared, i got the energy.
04:53
In this case, one c square.
04:56
That gives me the energy.
04:59
So this becomes, when you think of the c squares as canceling out, this becomes 4gv over 5, g .e .v, square that, one half.
05:16
And this comes out to be 0 .60c.
05:22
So that's at speed.
05:24
Oh, i gotta put a c there...