$37.$ A ball moving with horizontal velocity $\backslash (\backslash$mathrm$\{$v$\}\_\{0\}\backslash )$ drops from A to a frictionless floor. Knowing that the ball hits the floor at a distance $5$ in from $\backslash ($ B $\backslash )$ and the height of the first bounce is $16$ in$.,$ determine $($a$)$ the coefficient of restitution between the ball and the floor, and $($b$)$ the length $\backslash (\backslash$mathrm$\{$d$\}\_\{1\}\backslash )$ of the first bounce.