Two projectiles are thrown with same initial velocity making an angle of $45^{\circ}$ and $30^{\circ}$ with the horizontal respectively. The ratio of their respective ranges will be :

<p>Here's how to determine the ratio of the ranges for the two projectiles:</p> <p><strong>Understanding the Concepts</strong></p> <ul> <li><strong>Projectile Motion:</strong> Projectile motion is the motion of an object thrown or projected into the air, subject to only the acceleration of gravity. The object is called a projectile, and its path is called its trajectory.</li><br> <li><strong>Range:</strong> The range of a projectile is the horizontal distance it covers from its launch point to the point where it returns to the same vertical height.</li> </ul> <p><strong>Key Formula</strong></p> <p>The formula for the range (R) of a projectile is:</p> <p>$R = \frac{u^2 \sin 2\theta}{g}$</p> <p>where:</p> <ul> <li> R = Range</li><br> <li> u = Initial velocity (same for both projectiles)</li><br> <li> θ = Angle of projection</li><br> <li> g = Acceleration due to gravity (constant)</li> </ul> <p><strong>Calculations</strong></p> <ol> <li><strong>Projectile 1 (45° angle):</strong></li> </ol> <p>Let the range of the projectile launched at 45° be R₁.</p> <p>$$R_1 = \frac{u^2 \sin (2 \times 45^{\circ})}{g} = \frac{u^2 \sin 90^{\circ}}{g} = \frac{u^2}{g} $$</p> <ol> <li><strong>Projectile 2 (30° angle):</strong></li> </ol> <p>Let the range of the projectile launched at 30° be R₂.</p> <p>$$R_2 = \frac{u^2 \sin (2 \times 30^{\circ})}{g} = \frac{u^2 \sin 60^{\circ}}{g} = \frac{u^2 \sqrt{3}}{2g}$$</p> <ol> <li><strong>Ratio of Ranges (R₁ : R₂):</strong></li> </ol> <p>Divide the range of projectile 1 by the range of projectile 2:</p> <p>$$\frac{R_1}{R_2} = \frac{\frac{u^2}{g}}{\frac{u^2 \sqrt{3}}{2g}} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}$$</p> <p>To simplify the ratio, multiply both numerator and denominator by √3:</p> <p>$$\frac{R_1}{R_2} = \frac{2\sqrt{3} \times \sqrt{3}}{3 \times \sqrt{3}} = \frac{6}{3\sqrt{3}} = \frac{2}{\sqrt{3}}$$</p> <p>Therefore, the ratio of the ranges of the two projectiles is <strong>2 : √3</strong>, which corresponds to <strong>Option C</strong>.</p> <p></p>