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
$\therefore R_1=\frac{u^2\sin \left[2\left(45^{\circ }-\theta \right)\right]}{g}=\frac{u^2\sin \left(90^{\circ }-2\theta \right)}{g}$
$=\frac{u^2\cos 2\theta }{g}$
For angle of projection $\left(45^{\circ }+\theta \right)$, the horizontal range is
$R_2=\frac{u^2\sin \left[2\left(45^{\circ }+\theta \right)\right]}{g}=\frac{u^2\sin \left(90^{\circ }+2\theta \right)}{g}$
$=\frac{u^2\cos 2\theta }{g}$
$\therefore \frac{R_1}{R_2}=\frac{u^2\cos 2\theta /g}{u^2\cos 2\theta /g}=\frac{1}{1}$
$\therefore$ The range is the same.