Problem $1.$ The two balls A and $B,$ each of mass $m=1lbm,$ hang on two different strings, each of length $L=3ft,$ as shown. Ball $A$ is released from rest at an angle of ${}_A=35$ to the vertical. During its motion, it will eventually hit ball $B.$ The coefficient of restitution for the impact of the two balls is given by $e=0\mathrm{.}9.$

a$)$ Employ energy principles to determine the speed $v_A$ of ball A immediately before impact.

b$)$ Employ linear momentum principles to determine the speed $v_B^{\text{'}}$ of ball $B$ immediately after impact.

c$)$ Employ energy principles to determine the maximum angle ${}_{B,\text{m}\text{a}\text{x}}^{\text{'}}($to the vertical$)$ that the ball $B$ can swing after impact.

Hint: Is $I=32\mathrm{.}2f\frac{t}{s^2}$ as the gravitational acceleration.

Problem $3.$ The two propeller planes A and $B$ are flying in horizontal flight, as shown in Fig. $2.$ At the instant shown, plane A located at position $C$ has a speed of $v_A=64\frac{m}{s},$ and plane $B$ has a speed of $v_B=100\frac{m}{s}.$ While flying along the path CDEF $($with a circular path between $E$ and $F),$ plane $A$ is increasing its speed at a uniform rate to reach a speed of $88\frac{m}{s}$ when it arrives at position $F.$ Plane $B,$ on the other hand, decreases its speed at a uniform rate of $a_B=2\frac{m}{s^2}$ while flying eastward $($i$.$e$.$ in the positive $i-$direction$).$ For the sake of simplicity, it can be assumed that aerodynamic drag is negligible.

a$)$ Determine the tangential acceleration $a_t$ of plane $A.$

b$)$ Use the result found in Part a$)$ to determine both the speetd $v_A$ of plane $A$ at the instant when it reaches position $D$ and the time $t_{CD}$ that it takes for plane $A$ to fly from $C$ to $D.$

c$)$ Use the result found in Part b$)$ to determine the speed $v_B$ of plane $B$ at the instant when plane A reaches position $D.$

d$)$ Use the results found in Parts b$)$ and c$)$ to determine the velocity $($vector$).v_{\frac{A}{B}}$ of plane $A$ as seen from the pilot in plane $B$ at the instant when plane $A$ is at position $D.$

e$)$ Use the result found in Part a$)$ to determine the acceleration $($vector$)a_{\frac{A}{B}}$ of plane $A$ as seen from the pilot in plane $B$ at the instant when plane $A$ is at position $F.$

Figure $2$: Planes A and $B$ flying on different horizontal paths.

MECH&AE $102$

Problem $3.$ The small motor $M$ is pivoted about the $x-$axis through $O$ and gives its shaft $OA$ a constant angular speed $p$ in the direction shown relative to its housing. The entire unit is then set into rotation about the vertical $Z-$axis at the constant angular velocity $.$ Simultaneously, the motor pivots about the $x-$axis at the constant rate ${}^{}$ for an interval of motion. Determine the velocity $v_A$ and the acceleration $a_A$ of the center $A$ of the ball tool in terms of $.$