Galileo writes that for angles of projection | vedclass

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(C) The horizontal range $R$ of a projectile is given by the formula $R = \frac{u^2 \sin(2\alpha)}{g}$,where $u$ is the initial velocity,$\alpha$ is t

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(C) The horizontal range $R$ of a projectile is given by the formula $R = \frac{u^2 \sin(2\alpha)}{g}$,where $u$ is the initial velocity,$\alpha$ is the angle of projection,and $g$ is the acceleration due to gravity.For the first angle of projection $\alpha_1 = (45^\circ - 	heta)$:$R_1 = \frac{u^2 \sin(2(45^\circ - 	heta))}{g} = \frac{u^2 \sin(90^\circ - 2	heta)}{g} = \frac{u^2 \cos(2	heta)}{g}$.For the second angle of projection $\alpha_2 = (45^\circ + 	heta)$:$R_2 = \frac{u^2 \sin(2(45^\circ + 	heta))}{g} = \frac{u^2 \sin(90^\circ + 2	heta)}{g} = \frac{u^2 \cos(2	heta)}{g}$.Comparing the two ranges,we see that $R_1 = R_2$.Therefore,the ratio of the horizontal ranges is $R_1 : R_2 = 1 : 1$.

Galileo writes that for angles of projection of a projectile at angles $(45^\circ + \theta)$ and $(45^\circ - \theta)$,the horizontal ranges described by the projectile are in the ratio of (if $\theta \le 45^\circ$):

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