Two projectiles are thrown with same initial velocity making an angle of $$45°$$ and $$30°$$ with the horizontal respectively. The ratio of their respective ranges will be

We need to find the ratio of ranges of two projectiles thrown with the same initial velocity at angles $$45°$$ and $$30°$$ with the horizontal.

Since the range of a projectile is given by $$R = \dfrac{u^2 \sin 2\theta}{g}$$, substituting $$\theta = 45°$$ gives $$R_1 = \dfrac{u^2 \sin(2 \times 45°)}{g} = \dfrac{u^2 \sin 90°}{g} = \dfrac{u^2}{g}$$.

Next, for $$\theta = 30°$$ we get $$R_2 = \dfrac{u^2 \sin(2 \times 30°)}{g} = \dfrac{u^2 \sin 60°}{g} = \dfrac{u^2 \sqrt{3}}{2g}$$.

From this, $$\dfrac{R_1}{R_2} = \dfrac{\dfrac{u^2}{g}}{\dfrac{u^2 \sqrt{3}}{2g}} = \dfrac{u^2}{g} \times \dfrac{2g}{u^2 \sqrt{3}} = \dfrac{2}{\sqrt{3}}$$.

Therefore, $$R_1 : R_2 = 2 : \sqrt{3}$$.

The correct answer is Option C: $$2 : \sqrt{3}$$.