$(10$ 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 $y-$direction with speed $v_1$

Particle $2$ is traveling in the negative $y-$direction with speed $v_2$

Particle $3$ is traveling in the positive $x-$direction with speed $v_3$

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 $\frac{v_1}{v_2}.$ 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 $v.$

$($d$)$ Find the ratio v$3\frac{?}{v}.$ You should be able to get a number or a fraction.