A $0\mathrm{.}5-$kilogram block slides at $\backslash (20\backslash$mathrm$\{$~m$\}/\backslash$mathrm$\{$s$\}\backslash )$ on a smooth frictionless surface toward a stationary sphere. The sphere is half the volume of the block but is eight times as dense. The block strikes the sphere at time $\backslash (\backslash$mathrm$\{$t$\}=0\backslash ).$ A plot of the force exerted on the cube by the ball as a function of time is shown. Determine the

$($a$)$ impulse applied to the block.

$[2$ points$]$

$($b$)$ speed of the ball immediately following the collision.

$[3$ points$]$

$($c$)$ direction and magnitude of the velocity of the cube immediately following the collision.

$[2$ points$]$