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Galileo, in his book Two new sciences, stated that “for elevations which exceed or fall short of $45^o$ by equal amounts, the ranges are equal”. Prove this statement.
Solution
Answer For a projectile launched with velocity
$v _{ o }$ at an angle $\theta_{ o },$ the range is given by
$R=\frac{v_{o}^{2} \sin 2 \theta_{0}}{g}$
Now, for angles, $\left(45^{\circ}+\alpha\right)$ and $\left(45^{\circ}-\alpha\right), 2 \theta_{0}$ is $\left(90^{\circ}+2 \alpha\right)$ and $\left(90^{\circ}-2 \alpha\right),$ respectively. The values of $\sin \left(90^{\circ}+2 \alpha\right)$ and $\sin \left(90^{\circ}-2 \alpha\right)$ are
the same, equal to that of $\cos 2 \alpha .$ Therefore, ranges are equal for elevations which exceed or fall short of $45^{\circ}$ by equal amounts $\alpha$
Similar Questions
Given that $u_x=$ horizontal component of initial velocity of a projectile, $u_y=$ vertical component of initial velocity, $R=$ horizontal range, $T=$ time of flight and $H=$ maximum height of projectile. Now match the following two columns.
| Column $I$ | Column $II$ |
| $(A)$ $u_x$ is doubled, $u_y$ is halved | $(p)$ $H$ will remain unchanged |
| $(B)$ $u_y$ is doubled $u_x$ is halved | $(q)$ $R$ will remain unchanged |
| $(C)$ $u_x$ and $u_y$ both are doubled | $(r)$ $R$ will become four times |
| $(D)$ Only $u_y$ is doubled | $(s)$ $H$ will become four times |
