$9.(10$ points$)$ High energy particles can be created in particle accelerators like the Large Hadron Collider. These high energy particles are not in their natural state and they will sometimes spontaneously decay $($split$)$ into many new particles. Consider such a spontaneous decay where a particle of mass m and speed v is traveling in the positive x $-$direction. All of the sudden it splits into $3$ particles.

$-$ Particle $1$ is traveling in the positive $\backslash ($ y $\backslash )-$direction with speed $\backslash (\backslash$mathbf$\{$v$\}\_\{1\}\backslash )$

$-$ Particle $2$ is traveling in the negative $\backslash ($ y $\backslash )-$direction with speed $\backslash (\backslash$mathbf$\{$v$\}\_\{2\}\backslash )$

$-$ Particle $3$ is traveling in the positive x $-$direction with speed $\backslash (\backslash$mathbf$\{$v$\}\_\{3\}\backslash )$

$-$ Particle $2$ is twice as massive as particle $1.$

$-$ Particle $3$ is three times as massive as particle $1.$

$($a$)$ Do you expect particle $1$ to be traveling faster or slower than particle $2?$ Use proportional reasoning sense$-$making to determine which should be traveling faster.

$($b$)$ Find the ratio $\backslash (\backslash$mathbf$\{$v$\}\_\{1\}/\backslash$mathbf$\{$v$\}\_\{\backslash$mathbf$\{2\}\}\backslash ).$ You should be able to get a number or a fraction.

$($c$)$ Do you expect particle $3$ to be traveling faster or slower than the original high energy particle? Use proportional reasoning sense$-$making to determine whether it should be traveling faster or slower than the original particle's speed $\backslash (\backslash$mathbf$\{$v$\}\backslash ).$

$($d$)$ Find the ratio $\backslash ($ v$\_\{3\}/\backslash$mathbf$\{$v$\}\backslash ).$ You should be able to get a number or a fraction.