Question
Galileo, in his book Two New Sciences, stated the “for elevations which exceed or fall short of 45° by equal amounts, the ranges are equal.” Prove this statement.
$\theta=45^\circ+\alpha.$
Range, $\text{R}_1=\frac{\text{u}^2\sin2(45^\circ+\alpha)}{\text{g}}$
$=\frac{\text{u}^2\cos2\alpha}{\text{g}}$
Case II: When angle of projection.
$\theta=45^\circ-\alpha.$
Range, $\text{R}_2=\frac{\text{u}^2\sin2(45^\circ-\alpha)}{\text{g}}$
$=\frac{\text{u}^2}{\text{g}}\sin(90^\circ-2\alpha)$
$=\frac{\text{u}^2}{\text{g}}\cos^2\alpha=\text{R}_1$
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