Relativity Question: Solving for E, p, KE and v/c

[kidsmoker]

Homework Statement

http://img140.imageshack.us/img140/1650/25406466.jpg

Homework Equations

$$E^{2}= (pc)^{2}+m_{0}^{2}c^{4}$$

The Attempt at a Solution

To find the energy I said that in the rest frame of the pion, it has energy 139.6Mev, so this has to equal the sum of the energies of the decay products. I then used conservation of momentum to find an equation for the energy of the muon:

$$E_{1}=\frac{139.6^2+105.7^2}{2*139.6} Mev = 109.8 Mev$$.

And so the energy of the neutrino is

$$E_{2}=139.6-109.8 Mev = 29.8 Mev$$.

I think these are correct. You can the rest energies from these total energies to get the kinetic energies of each:

$$KE_{1} = 109.8-105.7 Mev = 4.1 Mev$$
$$KE_{2} = 29.8 Mev$$.

The momentum can be calculated by rearranging the equation for $$E^{2}$$ given above, and I found

$$p_{1} = 29.7 Mev/c$$
$$p_{2} = 29.8 Mev/c$$ .

Hopefully those are all okay. I'm unsure as to how to find the speed with respect to c though. If I rearrange the equation for kinetic energy in order to find v, what is this v with respect to?

Thanks.

 

[anathema]

Dear forum post author,

Your approach to finding the energies and momenta of the decay products is correct. To find the speed of the decay products with respect to the speed of light, you can use the equation for kinetic energy:

KE = (γ - 1)mc^2

where γ is the Lorentz factor given by:

γ = 1/√(1 - v^2/c^2)

You can rearrange this equation to solve for v, which will give you the speed of the decay products with respect to the speed of light. Keep in mind that this speed may be different for each decay product, as they may have different kinetic energies.

I hope this helps! Let me know if you have any further questions.

 

[nlightner]

Your calculations for energy and momentum seem to be correct. To find the velocity with respect to the speed of light, you can use the equation v = p/E, where p is the momentum and E is the energy. This will give you the velocity in units of c. Alternatively, you could use the equation v = √(1-(m0c^2/E)^2) to find the velocity in terms of the speed of light.