Problem $3(40$ Points$)$

Sphere $A,$ which has a mass of $m_A=0\mathrm{.}5kg,$ is initially at rest as it hangs from a cord when it collides with block $B,$ which has a mass of $m_B=1kg$ and is traveling to the left at $v_{B_1}=2\frac{m}{s}.$

a$.)$ If the impact between sphere A and block $B$ is a central impact with a coefficient of restitution of $e=0\mathrm{.}8,$ determine the speeds $v_{A_2}$ and $v_{B_2}$ of sphere A and block $B$ after impact.

b$.)$ What fraction of the kinetic energy of the system before the impact is lost due to the impact?

c$.)$ If no non$-$conservative forces act on sphere $A$ as it swings after the impact, determine the maximum height $h$ it reaches above the ground, where it is momentarily at rest at this height.

d$.)$ If block $B$ slides along a rough surface with a coefficient of friction of ${}_k=0\mathrm{.}60$ after the impact, determine the distance $x$ it travels before coming to a stop.

e$.)$ If block $B$ was originally launched from a spring with a stiffness of $k=5000\frac{N}{m}$ located $1$ m away from the impact point, what compression $s$ of the spring is needed for block $B$ to reach a speed of $v_{B_1}=2\frac{m}{s}$ at the impact point if it travels over the same rough surface with ${}_k=0\mathrm{.}60$ before hitting sphere $A?$