$29.$ Among other things, Mr$.$ Chevalier is an excellent pool player. His expertise in trick shots is due to his vast understanding of physics. The billiard balls are set$-$up as seen in the figure to the right. Mr$.$ Chevalier hits the cue ball with an initial velocity of $\backslash (2\mathrm{.}8\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash )$ and it collides with a stationary $8-$ball which moves forward to collide with the stationary $2-$ball and $3-$ball. Each billiard ball has an equal mass. Assume friction is negligible.

a$)$ Determine the velocity of the $8-$ball and cue ball after the first collision. $[$A$-4]$

b$)$ Prove that the first collision is elastic. $[$A$]$

c$)$ Draw a diagram that represents the $\backslash (2^\{\backslash$text $\{$nd $\}\}\backslash )$ collision. $[$A$]$

d$)$ The $8-$ball continues on its path and hits the $2-$ball and $3-$ball at the same time and then continues forward with a velocity of $\backslash (0\mathrm{.}50\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash ).$ The collision causes both balls to deflect at the same speed with an angle of $\backslash (30^\{\backslash$circ$\}\backslash )$ from the original direction. Mr$.$ Chevalier knows that friction will affect his shot and calculates that the minimum speed must ve at ieast i$.$cv ins. $$uvvevei$,$ ue speed cannot exceed $\backslash (1\mathrm{.}30\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash )$ or the billiard balls with bounce out of the corner pockets. Does Mr$.$ Chevalier actually make the trick shot? $[$T $-4]$