Problem $2(10$ pts$)$

Refer to Fig. $2$: a two$-$bladed aircraft propeller follows a circular path of radius $R(R$ does not change over time$)$ at an altitude $h,$ which changes over time. A brave spider $s$ crawls along one of the blades at a constant speed $v.$

$($a$)$ Determine the velocity of the spider with respect to the airplane. Express it in terms of the airplane frame $A\{hat(a{)}_1,$hat$(a{)}_2,$hat$(a{)}_3\}.(3$ pts$)$

$($b$)$ Determine the velocity of the airplane with respect to ground, i$.$e$.$ with respect to the inertial frame $E\{hat(e{)}_1,$hat$(e{)}_2,$hat$(e{)}_3\}.$ Express it in terms of the $E$ frame. $(3$ pts$)$

$($c$)$ Determine the velocity of the spider with respect to ground, i$.$e$.$ with respect to the inertial frame $E\{hat(e{)}_1,$hat$(e{)}_2,$hat$(e{)}_3\}.$ Express it in terms of the airplane frame, $A.(3$ pts$)$

$($d$)$ How long it takes for the spider to reach the end of the blade of length $l$ if it starts at $q$ and has a velocity of $v?(1$ pt$)$

Fig. $2$: $E\{hat(e{)}_1,$hat$(e{)}_2,$hat$(e{)}_3\}$ : inertial frame. $A\{hat(a{)}_1,$hat$(a{)}_2,$hat$(a{)}_3\}$ : airplane frame. $B\{hat(b{)}_1,$hat$(b{)}_2,$hat$(b{)}_3\}$ : blades' frame.