Projectile Motion: Angle for Equal Height and Range

Question

What is the value of angle of projection for which the height of the projectile (H) and the range (R)of the projectile are the same?

Solution: Height and range

Maximum height of the projectile (H)

Height of the projectile (H)

$${H=\frac{u^2 \sin^2\theta}{2g}}$$

Range of the projectile

$${R=\frac{u^2\sin 2 \theta}{g}}$$

$${H=R}$$

$${\frac{u^2\sin^2\theta}{2g}=\frac{u^2\sin 2\theta}{g}}$$

$${\frac{\sin^2\theta}{2}=2\sin\theta\cos\theta}$$

$${\frac{\sin\theta}{\cos\theta}=4}$$

$${\tan\theta=4}$$

$${\theta=\tan^{-1} (4)}$$

$${\theta=76^\circ}$$

Range of the projectile

Projectile motion: