Trajectory of A Projectile - Angle Required To Hit Coordinate (x,y)

Angle Required To Hit Coordinate (x,y)

To hit a target at range x and altitude y when fired from (0,0) and with initial speed v the required angle(s) of launch are:

The two roots of the equation correspond to the two possible launch angles, so long as they aren't imaginary, in which case the initial speed is not great enough to reach the point (x,y) you have selected. The greatest feature of this formula is that it allows you to find the angle of launch needed without the restriction of y = 0.

Derivation

First, two elementary formulae are called upon relating to projectile motion:

(1)

(2)

Solving (1) for t and substituting this expression in (2) gives:

(2a)

(2b) (Trigonometric identity)

(2c) (Trigonometric identity)

(2d) (Algebra)

Let

(2e) (Substitution)

(2f) (Quadratic formula)

(2f) (Algebra)

(2g) (Substitution)

(2h) (Algebra)

Also, if instead of a coordinate (x,y) you're interested in hitting a target at distance r and angle of elevation (polar coordinates), use the relationships and and substitute to get: