Question

Two projectiles \(A\) and \(B\) are launched with the same speed at angles \(15^\circ\) and \(30^\circ\) respectively. Find the ratio of range \(A\) to range \(B\).

• A. \( \frac{2}{\sqrt{3}} \)
• B. \( \frac{1}{\sqrt{3}} \)
• C. \( \sqrt{3} \)
• D. \( 2\sqrt{3} \)

Answer 1

The Correct Option is B

Concept:
The horizontal range of a projectile launched with speed \(u\) at an angle \(\theta\) with the horizontal is given by \[ R = \frac{u^2 \sin 2\theta}{g} \] Thus, when the initial speeds are the same, the ratio of ranges depends only on \(\sin 2\theta\). Step 1: Write the range expressions for both projectiles. \[ R_A = \frac{u^2 \sin 2\theta_A}{g} \] \[ R_B = \frac{u^2 \sin 2\theta_B}{g} \] Step 2: Take the ratio of ranges. \[ \frac{R_A}{R_B} = \frac{\frac{u^2 \sin 2\theta_A}{g}} {\frac{u^2 \sin 2\theta_B}{g}} \] \[ \frac{R_A}{R_B} = \frac{\sin 2\theta_A}{\sin 2\theta_B} \] Step 3: Substitute the given angles. \[ \theta_A = 15^\circ, \qquad \theta_B = 30^\circ \] \[ \frac{R_A}{R_B} = \frac{\sin 30^\circ}{\sin 60^\circ} \] \[ = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} \] \[ \frac{R_A}{R_B} = \frac{1}{\sqrt{3}} \]