Question

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

• A. \(\sqrt{2}\)
• B. \(\sqrt{3}\)
• C. \(2\sqrt{5}\)
• D. \(\frac{1}{\sqrt{2}}\)

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The horizontal range (\(R\)) of a projectile depends on the initial velocity \(u\) and the angle of projection \(\theta\). We need to compare the ranges for two different angles with the same velocity.

Step 2: Key Formula or Approach:
1. Horizontal Range \(R = \frac{u^2 \sin(2\theta)}{g}\). 2. Since \(u\) and \(g\) are constant, \(R \propto \sin(2\theta)\).

Step 3: Detailed Explanation:
1. For \(\theta_1 = 15^{\circ}\): - \(R_1 \propto \sin(2 \times 15^{\circ}) = \sin(30^{\circ}) = 1/2\). 2. For \(\theta_2 = 30^{\circ}\): - \(R_2 \propto \sin(2 \times 30^{\circ}) = \sin(60^{\circ}) = \sqrt{3}/2\). 3. Ratio \(R_1 : R_2 = \frac{1/2}{\sqrt{3}/2} = 1 : \sqrt{3}\). 4. Given ratio is \(1:x\), so \(x = \sqrt{3}\).

Step 4: Final Answer:
The value of \(x\) is \(\sqrt{3}\).