- #1

lriuui0x0

- 101

- 25

- Homework Statement
- A particle ##P## of mass ##M## decays into a particle ##R## with mass ##0 < m < M## and a massless particle. Calculate the speed of ##R## in ##P##'s rest frame

- Relevant Equations
- ##m^2c^2 = E^2/c^2 - \mathbf{p}^2##

I have attempted a solution using conservation of momentum. Could people help check if this solution is correct (the result looks weird), as the problem doesn't have solution with it.

$$

\begin{aligned}

\begin{pmatrix}Mc \\ 0\end{pmatrix} &= \begin{pmatrix}E_R/c \\ \mathbf{p}_R\end{pmatrix} + \begin{pmatrix}E_\gamma/c \\ \mathbf{p}_\gamma \end{pmatrix} \\

\begin{pmatrix}Mc \\ 0\end{pmatrix} &= \begin{pmatrix}\sqrt{m^2c^2 + \mathbf{p}_R^2} \\ \mathbf{p}_R\end{pmatrix} + \begin{pmatrix}|\mathbf{p}_R| \\ -\mathbf{p}_R \end{pmatrix} \\

M^2c^2 &= m^2c^2 + 2(\sqrt{m^2c^2 + \mathbf{p}_R^2}|\mathbf{p}_R| + \mathbf{p}_R^2) \\

M^2c^2 &= m^2c^2 + 2(m\gamma v\sqrt{m^2c^2 + m^2\gamma^2v^2} + m^2\gamma^2v^2) \\

M^2c^2 &= m^2c^2 + 2(m^2\gamma cv\sqrt{1 + \gamma^2(1-\gamma^{-2})} + m^2\gamma^2v^2) \\

M^2c^2 &= m^2c^2 + 2(m^2\gamma^2cv + m^2\gamma^2v^2) \\

M^2c^2 &= m^2(c^2 + 2\gamma^2cv + \gamma^2v^2) \\

M^2c^2 &= m^2(c^2 + \frac{2cv}{1-\frac{v^2}{c^2}} + \frac{v^2}{1-\frac{v^2}{c^2}}) \\

M^2c^2 &= m^2\frac{c^4 - c^2v^2 + 2c^3v + c^2v^2}{c^2 - v^2} \\

M^2 &= m^2\frac{c^2 + 2cv}{c^2 - v^2} \\

M^2c^2 - M^2v^2 &= m^2c^2 + 2m^2cv \\

M^2v^2 + 2m^2cv + (m^2-M^2)c^2 &= 0 \\

v^2 + 2r^2cv +(r^2-1)c^2 &= 0 \\

v &= (\sqrt{r^4-r^2+1} - r^2)c \\

\end{aligned}

$$

where we have replaced ##r = \frac{m}{M}##.

$$

\begin{aligned}

\begin{pmatrix}Mc \\ 0\end{pmatrix} &= \begin{pmatrix}E_R/c \\ \mathbf{p}_R\end{pmatrix} + \begin{pmatrix}E_\gamma/c \\ \mathbf{p}_\gamma \end{pmatrix} \\

\begin{pmatrix}Mc \\ 0\end{pmatrix} &= \begin{pmatrix}\sqrt{m^2c^2 + \mathbf{p}_R^2} \\ \mathbf{p}_R\end{pmatrix} + \begin{pmatrix}|\mathbf{p}_R| \\ -\mathbf{p}_R \end{pmatrix} \\

M^2c^2 &= m^2c^2 + 2(\sqrt{m^2c^2 + \mathbf{p}_R^2}|\mathbf{p}_R| + \mathbf{p}_R^2) \\

M^2c^2 &= m^2c^2 + 2(m\gamma v\sqrt{m^2c^2 + m^2\gamma^2v^2} + m^2\gamma^2v^2) \\

M^2c^2 &= m^2c^2 + 2(m^2\gamma cv\sqrt{1 + \gamma^2(1-\gamma^{-2})} + m^2\gamma^2v^2) \\

M^2c^2 &= m^2c^2 + 2(m^2\gamma^2cv + m^2\gamma^2v^2) \\

M^2c^2 &= m^2(c^2 + 2\gamma^2cv + \gamma^2v^2) \\

M^2c^2 &= m^2(c^2 + \frac{2cv}{1-\frac{v^2}{c^2}} + \frac{v^2}{1-\frac{v^2}{c^2}}) \\

M^2c^2 &= m^2\frac{c^4 - c^2v^2 + 2c^3v + c^2v^2}{c^2 - v^2} \\

M^2 &= m^2\frac{c^2 + 2cv}{c^2 - v^2} \\

M^2c^2 - M^2v^2 &= m^2c^2 + 2m^2cv \\

M^2v^2 + 2m^2cv + (m^2-M^2)c^2 &= 0 \\

v^2 + 2r^2cv +(r^2-1)c^2 &= 0 \\

v &= (\sqrt{r^4-r^2+1} - r^2)c \\

\end{aligned}

$$

where we have replaced ##r = \frac{m}{M}##.