Problem $4$: Block $A($with mass $10$ kg $)$ is pushed up against a spring, compressing it a distance $x=0\mathrm{.}1$ m $.$ The block is then released from rest and slides down the $20$ incline until it strikes sphere $B($with mass $2$ kg $)$ that is suspended by a rope of length $L=1\mathrm{.}5$ meters. The spring constant is $k=1500\frac{N}{m}.$ The coefficient of friction ${}_k$ between A and the surface is $0\mathrm{.}15.$ The coefficient of restitution between A and $B$ is $0\mathrm{.}9.$ Block A slides a distance of $x+d=1$ meter. Pendulum $B$ reaches a highpoint at Position $3.($a$)$ Find the velocity of A just before it strikes pendulum bob $B.$

$($b$)$ If the coefficient of friction on the incline is increased significantly, the velocity of $A$ at Position $2$ becomes an imaginary number. Why?

$($c$)$ Block A hits $B.$ Use the principle of impulse and momentum to find the velocities after the impact. Show the Impulse$/$Momentum diagrams.

$($d$)$ Use conservation of energy to find the height of $B$ at Position $3.$