00:01
In this problem, we have the decay of radium to radon and a helium nucleus and alpha particle.
00:11
Now, what we need to do is look, the goal is to get the energy, kinetic energy, of the alpha particle, the helium nucleus.
00:22
In the previous problem that they were made reference to, you were told to ignore, say that the radon took none of the kinetic energy that was given to the decay products.
00:35
And the helium nucleus got it all.
00:38
Now we're going to remove that assumption or that restriction and see how it all plays out and what the helium nucleus gets in this case.
00:48
So we're going to need energy and momentum.
00:51
So let me write the energy conservation expression.
00:56
So the radium is only at rest, so it only has rest energy.
01:00
M radium c squared equals.
01:03
And then we have the rest energy of radon plus the kinetic energy, connect energy, radon plus the rest energy of the helium plus the kinetic energy of helium.
01:28
That's our energy expression.
01:32
Now, let me introduce this quantity q.
01:36
It tells me how much rest mass got converted to kinetic energy, effectively.
01:44
Rest energy got converted.
01:45
To connect energy.
01:47
And so it's defined in the following matter.
01:50
So you take the rest energy before, subtract the rest energy after.
01:55
Really, you're dealing with the rest energy is connected with the mass.
01:59
So you're dealing with what's happening to the mass.
02:01
There's any mass being converted from mass to connect energy.
02:08
So m -r -a -c -squared minus radon, c -squared plus helium.
02:21
C squared.
02:24
And this, if you bring all these terms to the left in the energy expression, and anything else, there could be kinetic energy on the left here too, but in our case there isn't.
02:35
Bring all the terms to the left, you're going to be left with kinetic energy of the radon, connect energy of the helium.
02:45
That's what we're left with.
02:46
That's what q is equal to.
02:49
So we actually made use of the energy expression in here.
02:55
The relationship between those, i should mention, if you're wondering going forward, can we use non -relativistic formulation? the answer is yes.
03:10
This quantity here, it's also the answer from the previous question, is 4 .8707 m .ev.
03:18
The rest energy of helium is 3 ,728 m .ev.
03:25
Whenever the kinetic energy is much less than the rest energy, then you can use non -relativistic.
03:32
Formulation.
03:37
And if you want to calculate all that out, i'll show you the conversion factors below, and you can justify what i said in terms of the numbers.
03:47
Now, let's do momentum.
03:51
We have zero, because radium is not moving.
03:54
P, radon, p helium.
04:01
And that tells me they're in opposite directions, but more important for our purposes, that the magnitudes are the same.
04:08
This is what i care about here.
04:10
The opposite direction is fine, but this is what matters to us.
04:15
Magnitude of the momentum for the radon is equal to the magnitude of the momentum of the upper particle.
04:24
Now, like i said, you also have the minus sign to.
04:27
I don't want to, this is all that's important for our purpose.
04:31
Let's square.
04:33
Let's square both sides.
04:36
Remember, connect energy and momentum non -voltavistically is related in this manner, p squared over 2m.
04:44
So prn squared, i'm going to be divided by something in a second.
04:48
Phe squared...