Answer
a) $\overline\omega_{max}=110.6rad/s$
b) $\overline\omega_{max}=134.2rad/s$
c) $\overline\omega_{max}=117.6rad/s$
Work Step by Step
The amount of time it takes for the entire arrow with length $L$ and speed $v$ to pass through the open space is $$t_{need}=\frac{L}{v}$$
We assume that for the arrow to completely pass through the open space, the propeller's angular speed cannot exceed a point $\overline\omega_{max}$ when it goes a distance $\Delta\theta=\pi/3 rad$ in time $t_{need}$. In other words, $$\overline\omega_{max}=\frac{\Delta\theta}{t_{need}}=\frac{\pi v}{3L}$$
a) $L=0.71m$ and $v=75m/s$
$$\overline\omega_{max}=\frac{\pi\times75}{3\times0.71}=110.6rad/s$$
b) $L=0.71m$ and $v=91m/s$
$$\overline\omega_{max}=\frac{\pi\times91}{3\times0.71}=134.2rad/s$$
c) $L=0.81m$ and $v=91m/s$
$$\overline\omega_{max}=\frac{\pi\times91}{3\times0.81}=117.6rad/s$$
