# Homework Help: Particle decays

1. Jun 4, 2009

### fredrick08

1. The problem statement, all variables and given/known data
a neutral kaon at rest decays to pion- and pion+ , if kaon rest mass is 497.7MeV/c^2, and the pions rest mass is 139.6MeV/c^2, what are the kinetic energies of the resulting pions?

Im very stuck with this, and not sure where to start, how can i find Ek of pions, if i dont know their velocities? ive searched thorugh my textbook but can find any similar examples, please someone help.

2. Jun 4, 2009

### Cyosis

Use conservation of energy and momentum.

3. Jun 4, 2009

### malawi_glenn

and conservation of momentum.

What are the relation between momentum and kinetic energy in relativistic dynamics?

4. Jun 4, 2009

### fredrick08

i dont need to use invarient mass E^2=(pc)^2+(mc^2)^2?

5. Jun 4, 2009

### Cyosis

Just write down the conservation equations in terms of $E_0=mc^2, \; E=\gamma mc^2, \; T=(\gamma-1)mc^2$.

6. Jun 4, 2009

### malawi_glenn

maybe, there are several ways to do this, i would just to what Cyosis suggested.

7. Jun 4, 2009

### fredrick08

so Ef=E1+E2=497.7MeV=.5mv^2+.5mv^2???? i cant do anything from there...

8. Jun 4, 2009

### Cyosis

You will have to use special relativity here. The total energy on one side equals the total energy on the other side. Total energy includes rest mass.

Edit: While your equation is wrong I don't see how you can say that you can't do anything from there. You already assumed that both speeds are equal (something you should show) so just solving for 1/2 m v^2 would give you the answer, albeit the wrong answer, because you're working classically.

9. Jun 4, 2009

### fredrick08

so 497.7MeV=mv^2+2mc^2??? im sorry i completely dont understand... could you tell me the formula im supposed to use, im unfamiliar with this relativistic part

10. Jun 4, 2009

### Cyosis

Where does this mv^2 come from. Relativistic kinetic energy is not 1/2 mv^2. In a collision we have three types of energies. Total energy, E, relativistic kinetic energy, T and rest energy E_0. Therefore we know that $E=T+E_0$. So what is the relativistic expression for kinetic energy?

Last edited: Jun 4, 2009
11. Jun 4, 2009

### fredrick08

so Ek=T-Eo? my book says Ek=mc^2(gamma-1)

12. Jun 4, 2009

### Cyosis

I defined T as the kinetic energy so no. If $E=T+E_0$ then $T=E-E_0$. You can call T E_k if you prefer that it does not matter as long as you don't mix them up.

13. Jun 4, 2009

### fredrick08

ok then T=gamma*mc^2-mc^2? but how do i find the gamma value, sine i dont know u?

14. Jun 4, 2009

### Cyosis

You are asked to find the kinetic energy not the velocity.

Can you please just write down the momentum conservation equation first, then the energy conservation equation?

15. Jun 4, 2009

### fredrick08

Pf=Pi? E/c=E/c+E/c and Ef=Ei

16. Jun 4, 2009

### Cyosis

A little bit more specific to this problem perhaps? What is the value of p-initial?

17. Jun 4, 2009

### fredrick08

p initial = 0, since the particle is at rest, therefore p final also has to equal 0?

18. Jun 4, 2009

### fredrick08

so are you saying that for the system total energy is m(k)c^2=2m(pi)c^2+2*gamma*m*u^2?

19. Jun 4, 2009

### Cyosis

You have way more information than that. You need to make use of it. I will list you all the variables you need to put in your equations.

$E_{kaon},T_{\pi^-},T_{\pi^+},E_{0,\pi^-},E_{0,\pi^+},p_{kaon},p_{\pi^-},p_{\pi^+}$

The first thing you do when solving a problem is writing down all the variables that are relevant to the problem. Now use these variables to write down the conservation of momentum equation and the conservation of energy equation.

20. Jun 4, 2009

### Cyosis

Where does the u come from? And secondly how do you know that both pions have the same kinetic energy (it's true)?

21. Jun 4, 2009

### fredrick08

if this is true, dont i need the gamma value?? but i need the speed of the pions to calculate that? but if they have a speed then they have a momentum. which doesnt make sense, since initial momentum is 0?

22. Jun 4, 2009

### Cyosis

Momentum is a vector, both pions do have non-zero momentum. From this you can derive that the kinetic energy of both pions is the same. Write it down in equation form using the variables I supplied.

23. Jun 4, 2009

### fredrick08

ok so the whole thing is...

Ekaon=Eo(pi-)+T(pi-)+Eo(pi+)+T(pi+)

and momentum pkaon=p(pi+)+p(pi-)

24. Jun 4, 2009

### Cyosis

Yep that is correct. You know that pkaon=0 so what is the relation between p(pi+) and p(pi-)?

25. Jun 4, 2009

### fredrick08

they must be equal.... and opposite?