Question

A gun is used to fire a bullet at an angle of \( 30^\circ \) and at \( 60^\circ \) respectively. The ratios of heights and ranges reached by the bullet are

• A. \( 1:1, 1:1 \)
• B. \( 1:3, 1:1 \)
• C. \( 3:1, 1:1 \)
• D. \( 1:1, 3:1 \)

Hint:

Angles \( \theta \) and \( 90^\circ - \theta \) give same range but different heights.

Answer 1

The Correct Option is B

Step 1: Write formula for maximum height.
\[ H = \frac{u^2 \sin^2\theta}{2g} \]

Step 2: Compare heights at \( 30^\circ \) and \( 60^\circ \).
\[ H_1 \propto \sin^2 30^\circ = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]
\[ H_2 \propto \sin^2 60^\circ = \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{3}{4} \]

Step 3: Ratio of heights.
\[ H_1 : H_2 = \frac{1}{4} : \frac{3}{4} = 1 : 3 \]

Step 4: Write formula for range.
\[ R = \frac{u^2 \sin 2\theta}{g} \]

Step 5: Compare ranges.
\[ R_1 \propto \sin 60^\circ = \frac{\sqrt{3}}{2} \]
\[ R_2 \propto \sin 120^\circ = \frac{\sqrt{3}}{2} \]

Step 6: Ratio of ranges.
\[ R_1 : R_2 = 1 : 1 \]

Step 7: Final Answer.
\[ \boxed{1:3, \; 1:1} \]