Question

For angles of projection of a projectile at angles $\left(45^{\circ}-\theta\right)$ and $\left(45^{\circ}+\theta\right)$, the horizontal ranges described by the projectile are in the ratio of:

• A. 2 : 1
• B. 1 : 1
• C. 2 : 3
• D. 1 : 2

Answer 1

Answer: (B)

Horizontal range $R=\frac{u^{2} \sin 2 \theta}{g}$
For angle of projection $\left(45^{\circ}-\theta\right)$, the horizontal range is
$\therefore \quad R_{1}=\frac{u^{2} \sin \left[2\left(45^{\circ}-\theta\right)\right]}{g}=\frac{u^{2} \sin \left(90^{\circ}-2 \theta\right)}{g} $
$=\frac{u^{2} \cos 2 \theta}{g}$
For angle of projection $\left(45^{\circ}+\theta\right)$, the horizontal range is
$R_{2}= \frac{u^{2} \sin \left[2\left(45^{\circ}+\theta\right)\right]}{g}=\frac{u^{2} \sin \left(90^{\circ}+2 \theta\right)}{g} $
$=\frac{u^{2} \cos 2 \theta}{g} $
$\therefore \quad \frac{R_{1}}{R_{2}}=\frac{u^{2} \cos 2 \theta / g}{u^{2} \cos 2 \theta / g}=\frac{1}{1}$
$\therefore$ The range is the same.