Solving Relatavistic Decay Homework

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Homework Statement

http://i45.tinypic.com/2it0cpy.jpg

Homework Equations

The Attempt at a Solution

momentum before is gamma*pionmass*0.93c
momentum after is gamma(u)*mass(u)*velocity(u) - E/c where E=pc which is the energy of massless particles

energy before is gamma*pionmass*c*c
energy after is gamma*mass(u)*c*c + pc

just wondering if what I've done above is correct as I've tried to do it that way and i get a really ridiculous answer. Any help is appreciated

btw the answer is (masspion)^2 + (mass(u)^2)/ 2(masspion).. anyone know how they got to that, its 110 MeV, after some extreme calculation i got 103.75 MeV but i don't know how they got it simply by doing a simple calculation as above
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[jbsteele]

The "really ridiculous answer" is probably due to the fact that you are adding your energies in terms of $c^2$, i.e. $E = mc^2$. This is incorrect; the energy $E$ should be the total energy of the particle, which is the rest mass energy plus the kinetic energy. The momentum $p$ is related to the kinetic energy by $E_{kin} = pc$.For this problem, the initial energy is $E_{i} = \gamma m_{\pi} c^2$, where $\gamma$ is the Lorentz factor for the pion's initial velocity. The momentum of the pion is $\vec{p} = \gamma m_{\pi} \vec{v}_{\pi,i}$. The final energy and momentum are given by\begin{align*}E_{f} &= \gamma m_{u} c^2 + pc \;\;\; (\text{where } c = \text{speed of light}) \\\vec{p}_{f} &= \gamma m_{u} \vec{v}_{u,f} - \frac{E_{kin}}{c} \;\;\; (\text{where } E_{kin} = pc)\end{align*}where $m_u$ is the mass of the particle after the decay and $\vec{v}_{u,f}$ is its final velocity. Note that $E_{kin}$ is the kinetic energy of the particle after the decay, not the initial kinetic energy of the pion.Now we can use the conservation of energy and momentum to solve for $m_u$. We have\begin{align*}E_{i} &= E_{f} \\\vec{p}_{i} &= \vec{p}_{f}\end{align*}Solving for $m_u$, we get\begin{align*}m_u &= \frac{E_{i}}{\gamma c^2} - \frac{\vec{p}_{i}\cdot\vec{v}_{u,f}}{\gamma

 

[kerimek]

I would first suggest double checking your calculations to ensure accuracy. It is also important to make sure that you are using the correct equations and units for the problem.

Without seeing your specific calculations, it is difficult to determine where you may have gone wrong. However, the answer provided in the problem is likely correct, as it is a standard result in particle physics. It is possible that you may have missed a step or made a calculation error.

In terms of the approach to solving the problem, it is important to consider the conservation of momentum and energy in the decay process. This means that the total momentum and energy before the decay must equal the total momentum and energy after the decay.

In this case, the initial momentum and energy are given by the pion mass and the final momentum and energy are given by the mass and velocity of the decay products. By setting these equal to each other and solving for the unknown mass, the result provided in the problem can be obtained.

I would suggest reviewing your calculations and possibly seeking assistance from a classmate or instructor if you are still having trouble. It is also important to understand the underlying concepts and equations used in solving the problem, rather than just trying to replicate a specific calculation.