The projectile motion equations

projectile equationsrange formulatime of flight formulatrajectory equationscannonball problem

Range, maximum height, time of flight and the two velocity components — the standard results for a projectile launched into still air.

Horizontal Velocity Component

v_x = v \cos\theta

Horizontal component of a projectile's launch velocity — the part of the speed that carries it downrange at a constant rate.

Vertical Velocity Component

v_y = v \sin\theta

Vertical component of a projectile's launch velocity — the part of the speed that fights gravity and sets the time aloft.

Projectile Time of Flight

T = \frac{2 v_0 \sin\theta}{g}

Total time a projectile stays airborne before returning to its launch height, set by launch speed and angle with g = 9.80665 m/s².

Projectile Maximum Height

H = \frac{v_0^{2} \sin^{2}\theta}{2g}

Peak height reached by a projectile launched at a given speed and angle above level ground, ignoring air resistance.

Projectile Range on Level Ground

R = \frac{v_0^{2} \sin 2\theta}{g}

Horizontal distance a projectile covers over level ground, from its launch speed and angle, ignoring air resistance.

Drop Height of a Horizontally Launched Projectile

y = \tfrac{1}{2} g t^{2}

Distance a horizontally launched projectile falls in a given time, independent of how fast it was thrown sideways.

How they fit together

Every one of these comes from a single idea: split the launch velocity into a horizontal part that never changes and a vertical part that obeys the kinematic equations with a = −g. Once you have v₀cos θ and v₀sin θ, time of flight is just the vertical problem, range is the horizontal speed multiplied by that time, and maximum height is where the vertical velocity reaches zero. Tartaglia worked out in 1537 that this makes 45° the best launch angle, and that 30° and 60° give the same range — one shot flat and fast, the other high and slow.

Choose by what the question hands you. A launch angle points to the component formulas first; a horizontal launch off a table or cliff is a different animal — there is no vertical component at all, so the fall time comes from the drop height alone and the horizontal speed never enters it. The two classic mistakes are computing sin θ where the range formula wants sin 2θ, and applying the level-ground range to a projectile that lands higher or lower than it started, which it simply does not cover. All of these also assume a vacuum: a real golf ball or shell falls well short.