Quantum Dynamics: Boron 14 Splitting into Carbon 14 and Electron

[thharrimw]

Homework Statement

a boron 14 nucleus (mass=14.02266u) is at rest splits into a electron (.00055U) and a carbon 14 (13.99995u) what are the speeds and KE for the carbon and the electron.

Homework Equations

i need help getting started on this so could someone help me get started.

The Attempt at a Solution

 

[neo86]

What you need is: conservation of momentum, conversation of energy (both together correspond to conservation of 4 Momentum)
then you need the relativistic Dispersion law:
E^2-\vec{p}^2=m^2
and the definition of kinetic energy:
E_{kin}=E-m
If you use all that and convert your units the right way you are done.

 

[thharrimw]

the only equations my book gave me are;
p=Ymu
E=Ymc^2
KE=Ymc^2-mc^2
KE=mc^2+1/2mu^2
deltaKE+delta mc^2
E^2=p^2C^2+m^2C^4
so i don't know if i could rewrite those equations to get the ones you told me to use. if i can can you show me how? if not can you show me how to solve the problem using those equations? I'm sorry if i seem like i don't know much but i have a quantum physics book and I'm trying to teach myself quantum physics.

 

[neo86]

Ok, I'll try to give a few hints:

You know that the boron was at rest, before the decay, that means by conservation of momentum:
\vec{p_C}=-\vec{p_e}
this means especially, that the momenta of carbon and electron have the same absolute value.
Since the energy is conserved (and we know the Boron was at rest, which means it had only it's rest Energy E=m_B c^2):
m_B c^2=E_e+E_C
now you plug in E^2=p^2c^2+m^2c^4 for e and C and use that the momentum for e and C hast the same absolute value(which I will denote by p):
m_B c^2 =2 p^2 c^2 +(m_e^2+m_C^2)c^4
Now you can use this to find p^2, since you now all the other quantities. then you plug this into:
p^2=\gamma^2 m^2 v^2
now you can plug in the formula for gamma and find v. (the speeds will be different for electron and Carbon since they have different masses).
To find E_{kin} you just plug p into E^2=p^2c^2+m^2c^4 for the electron and the Carbon, take the square root of it and substract m c^2 to get the kinetic energy

 

[thharrimw]

thanks that makes sense but what dose the arow above pc mean?

 

[neo86]

The arrow means that its a vector.

 

[thharrimw]

ok thanks that helped a lot