Galileo stated that the horizontal ranges of a projectile are equal for two specific angles of proje

Question

Galileo stated that the horizontal ranges of a projectile are equal for two specific angles of projection. If a projectile is launched with initial velocity $u$, for which of the following angle pairs will the range be identical?

Professor PC Thomas Classes · Answer

The horizontal range is $R = \frac{u^2\sin(2	heta)}{g}$. For angles $(45^\circ + \alpha)$ and $(45^\circ - \alpha)$, $2	heta$ becomes $(90^\circ + 2\alpha)$ and $(90^\circ - 2\alpha)$. Since $\sin(90^\circ + 2\alpha) = \sin(90^\circ - 2\alpha) = \cos(2\alpha)$, the ranges are equal.