Q126 / 180 · COMEDK 2025 (10 May Shift 2) · Physics
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
- Correct answerA$1: 3,1: 1$
- B$1: 1,1: 1$
- C$1: 1,3: 1$
- D$3: 1,1: 1$
Solution
Maximum height: $H = \dfrac{u^2 \sin^2\theta}{2g}$
$H_1 = \dfrac{u^2 \sin^2 30°}{2g} = \dfrac{u^2}{8g}$, $\quad H_2 = \dfrac{u^2 \sin^2 60°}{2g} = \dfrac{3u^2}{8g}$
$H_1 : H_2 = 1 : 3$
Range: $R = \dfrac{u^2 \sin 2\theta}{g}$
$R_1 = \dfrac{u^2 \sin 60°}{g}$, $\quad R_2 = \dfrac{u^2 \sin 120°}{g}$
Since $\sin 60° = \sin 120°$, $R_1 = R_2 \Rightarrow R_1 : R_2 = 1 : 1$
Ratio of heights : ranges $= 1:3,\ 1:1$
$H_1 = \dfrac{u^2 \sin^2 30°}{2g} = \dfrac{u^2}{8g}$, $\quad H_2 = \dfrac{u^2 \sin^2 60°}{2g} = \dfrac{3u^2}{8g}$
$H_1 : H_2 = 1 : 3$
Range: $R = \dfrac{u^2 \sin 2\theta}{g}$
$R_1 = \dfrac{u^2 \sin 60°}{g}$, $\quad R_2 = \dfrac{u^2 \sin 120°}{g}$
Since $\sin 60° = \sin 120°$, $R_1 = R_2 \Rightarrow R_1 : R_2 = 1 : 1$
Ratio of heights : ranges $= 1:3,\ 1:1$
Hence, the correct answer is option: 'A'.