Q.31. The two projectiles are projected with the same initial velocities at the 15° and 30° with respect to the horizontal. The ratio of their ranges is 1: x. The value of x is

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Correct answer: B. Sol. The range of a projectile is given by $R = \frac{u^2 \sin(2	heta)}{g}$. For the first projectile, $	heta_1 = 15^\circ$, so $R_1 = \frac{u^2 \sin(2 	imes 15^\circ)}{g} = \frac{u^2 \sin(30^\circ)}{g}$. For the second projectile, $	heta_2 = 30^\circ$, so $R_2 = \frac{u^2 \sin(2 	imes 30^\circ)}{g} = \frac{u^2 \sin(60^\circ)}{g}$. The ratio $\frac{R_1}{R_2} = \frac{\sin(30^\circ)}{\sin(60^\circ)} = \frac{1/2}{\sqrt{3}/2} = \frac{1}{\sqrt{3}}$. Given the ratio is $1:x$, so $\frac{1}{x} = \frac{1}{\sqrt{3}}$, which means $x = \sqrt{3}$.

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