Centripetal force $F = \frac{mv^2}{R} \propto \frac{1}{R^n}$.

• $v^2 \propto \frac{1}{R^{n-1}} \implies v \propto R^{(1-n)/2}$
• $T = \frac{2\pi R}{v} \propto \frac{R}{R^{(1-n)/2}} = R^{1 - (\frac{1-n}{2})} = R^{(n+1)/2}$.

The textbook explains the technical basis of cinema's illusion of motion: “Motion picture projectors present images much faster, at 24-frames per-second, with each of those frames flashed on the screen twice. This high speed makes the transition between images virtually invisible.” This means the human eye receives 48 flashes per second (24 frames × 2 flashes each), which is faster than the eye's ability to detect individual frames, creating the perception of smooth, continuous motion. The double-flash of each frame was introduced to reduce flicker — at 24 frames per second with a single flash, the lower light frequency caused visible flickering. Doubling the flash rate with a rotating shutter solved this problem. This 24fps standard was established in the early sound film era and remained the global cinema standard for most of the 20th century. Modern digital cinema can shoot at 48fps or higher.

For complementary angles ($60°$ and $30°$, since $60°+30°=90°$) with the same initial speed:

Range $R = \dfrac{u^2\sin2\theta}{g}$: $\sin(120°) = \sin(60°)$ ✓ — same range.

Time of flight $\propto \sin\theta$ — different. Max height $\propto \sin^2\theta$ — different.