Projectile Motion: From a Perfect Parabola to Real Ballistics

The clean case: no air, flat ground

Strip away air resistance and a launched object obeys the simplest possible motion: constant horizontal velocity, and constant downward acceleration g in the vertical direction. Split the launch velocity v0 at angle θ into components v0·cos θ and v0·sin θ, and the two directions decouple completely — gravity never touches the horizontal motion, and the horizontal motion never touches the vertical one. That independence is what turns the trajectory into an exact parabola, and it lets you write range, height and time of flight as closed-form formulas rather than something you have to simulate.

x(t) = v0 cos(θ) · t
y(t) = v0 sin(θ) · t − ½ g t²

time of flight   T = 2 v0 sin(θ) / g
max height       H = v0² sin²(θ) / (2g)
range            R = v0² sin(2θ) / g   ← maximised at θ = 45°

The range formula's dependence on sin(2θ) explains a fact every artillery table and every playground ball-thrower has stumbled on: 30° and 60° give exactly the same range, because sin(60°) = sin(120°). Only 45° sits at the peak, and it is the unique angle that gets you the farthest for a given launch speed — but only in this idealised, drag-free world.

Adding air resistance breaks the symmetry

Real projectiles push air out of the way, and that drag force always points opposite to the velocity vector, with a magnitude that grows with speed — quadratically for the turbulent flow regime most thrown or fired objects experience, F_drag = ½ ρ v² C_d A. Because drag depends on velocity, the horizontal and vertical equations of motion stop being independent: both now contain the total speed, so you can no longer solve them separately in closed form. The trajectory has to be stepped forward numerically, evaluating the drag force from the current velocity at each small time increment.

v = sqrt(vx² + vy²)
ax = −(k/m) · v · vx
ay = −g − (k/m) · v · vy      // k = ½ ρ C_d A, the drag coefficient bundle

Drag is fastest exactly where speed is highest — right after launch — so it robs the projectile of energy hardest at the start of the flight. The projectile reaches its peak later, slower and closer to the launch point than the no-drag formula would predict, then falls back down along a curve that is noticeably steeper than the angle it went up at. The once-symmetric parabola tilts into a lopsided arc, steep on the way down and comparatively shallow going up — the signature shape of every real ballistic trajectory, from a thrown baseball to an artillery shell.

Why 45 degrees is no longer optimal

Because drag punishes time spent in the air more than it punishes launch angle directly, the optimal angle for maximum range drops once air resistance is switched on — typically into the high 30s or low 40s of degrees, and lower still for objects with a large drag coefficient relative to their mass, like a beach ball compared with a shot put. A flatter launch trades some peak height for less time exposed to drag, and for a sufficiently draggy object that trade is worth making. Launching from an elevated position above the landing point pulls the optimal angle down further still, since gravity has extra distance to work with regardless of angle, while launching from below the landing point pushes it up. None of these effects show up in the flat-ground, no-drag formula — they only become visible once you actually integrate the equations of motion.

Changing the gravity: Earth, Moon, Jupiter

Because range scales as 1/g in the simple model, swapping the local gravitational acceleration rescales the whole trajectory. The Moon's gravity, about one sixth of Earth's, lets a projectile launched at the same speed and angle travel roughly six times as far and climb six times as high — the reason the Apollo astronauts' golf shots and long jumps looked so dramatically different from anything possible at home. Jupiter's gravity, about 2.5 times Earth's, does the opposite: it hauls a projectile back down long before it can build up any real range, flattening the same launch into a short, stubby arc.

Reading the range envelope

A useful way to see the whole family of possible trajectories for a fixed launch speed at once is the range envelope — the outer curve that bounds every trajectory you could reach by varying the angle. In the no-drag case that envelope is itself a parabola, called the parabola of safety in artillery contexts: it marks the farthest boundary any target-seeking angle can reach with a given muzzle speed, and any target inside it can be hit by exactly two angles that sum to 90°, while any target outside it is unreachable no matter what angle is chosen. Overlaying several launch angles side by side, as this simulation does from 15° to 75°, makes both the envelope and the 30°/60° symmetry visible at a glance.

Frequently asked questions

Try it live

Everything above runs in your browser — open Projectile Motion and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.