In a simplified model of a relativistic nucleus-nucleus collision, a nucleus of rest mass $m_{1}$ and speed $\beta_{1}$ collides head-on with a target nucleus of mass $m_{2}$ at rest. The composite system recoils at speed $\beta_{0}$ and with center of mass energy $\varepsilon_{0}$. Assume no new particles are created.

Calculate $\beta_{0}$ for a ${ }^{40} \mathrm{Ar}$ nucleus impinging at $\beta_{1}=0.8$ on a ${ }^{238} \mathrm{U}$ nucleus.
As implied by the question the velocity of light is taken to be one for convenience.

For a system, $E^{2}-p^{2}$ is invariant under Lorentz transformation. In the laboratory frame $\Sigma$, writing $\gamma_{1}=\frac{1}{\sqrt{1-\beta_{1}^{2}}}$,

$$
E^{2}-p^{2}=\left(m_{1} \gamma_{1}+m_{2}\right)^{2}-\left(m_{1} \gamma_{1} \beta_{1}\right)^{2} .
$$

In the center of mass frame $\Sigma^{\prime}, E^{\prime 2}-p^{\prime 2}=\varepsilon_{0}^{2}$. Hence

$$
\begin{aligned}
\varepsilon_{0}^{2} &=\left(m_{1} \gamma_{1}+m_{2}\right)^{2}-\left(m_{1} \gamma_{1} \beta_{1}\right)^{2} \\
&=m_{1}^{2} \gamma_{1}^{2}\left(1-\beta_{1}^{2}\right)+2 m_{1} m_{2} \gamma_{1}+m_{2}^{2} \\
&=m_{1}^{2}+m_{2}^{2}+2 m_{1} m_{2} \gamma_{1},
\end{aligned}
$$

or

$$
\varepsilon_{0}=\sqrt{m_{1}^{2}+m_{2}^{2}+\frac{2 m_{1} m_{2}}{\sqrt{1-\beta_{1}^{2}}}} .
$$

In the laboratory, the system of $m_{1}, m_{2}$ has total momentum $m_{1} \gamma_{1} \beta_{1}$ and total energy $m_{1} \gamma_{1}+m_{2}$. These are conserved quantities so that after the collision the composite system will move with velocity

$$
\beta_{0}=\frac{m_{1} \gamma_{1} \beta_{1}}{m_{1} \gamma_{1}+m_{2}}=\frac{m_{1} \beta_{1}}{m_{1}+m_{2} \sqrt{1-\beta_{1}^{2}}} .
$$

The masses are approximately

$$
\begin{aligned}
&m_{1}=40 \times 0.94=37.6 \mathrm{GeV}, \\
&m_{2}=238 \times 0.94=223.7 \mathrm{GeV} .
\end{aligned}
$$

Then

$$
\begin{aligned}
\varepsilon_{0} &=\sqrt{37.6^{2}+223.7^{2}+\frac{2 \times 37.6 \times 223.7}{\sqrt{1-0.64}}} \\
&=282 \mathrm{GeV}=2.82 \times 10^{5} \mathrm{MeV}, \\
\beta_{0} &=\frac{37.6 \times 0.8}{37.6+223.7 \times \sqrt{1-0.64}}=0.175
\end{aligned}
$$