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Motion in a Plane — Physics STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 SciencePhysicsMotion in a Plane1 Mark
Question
Galileo, in his book Two new sciences, stated that “for elevations which exceed or fall short of 45° by equal amounts, the ranges are equal”. Prove this statement.
✓

Answer

For a projectile launched with velocity $v _{ o }$ at an angle $\theta_{ o }$, the range is given by
$
R=\frac{v_0^2 \sin 2 \theta_0}{g}
$
Now, for angles, $(45+\alpha)$ and $(45-\alpha), 2 \theta_0$ is $(90+2 \alpha)$ and $(90-2 \alpha)$, respectively. The values of $\sin (90+2 \alpha)$ and $\sin (90-2 \alpha)$ are the same, equal to that of $\cos 2 \alpha$. Therefore, ranges are equal for elevations which exceed or fall short of 45 by equal amounts $\alpha$.

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