Projectile Motion Simulator — Interactive Physics Lab

How projectile motion works

A projectile is any object moving under gravity alone once it has been launched — a thrown ball, a launched shell, a jet of water. Its motion splits cleanly into two independent parts: horizontal motion at constant velocity, and vertical motion under constant gravitational acceleration. Those two axes do not affect each other, and that independence is the key insight that makes the problem solvable.

Because the horizontal velocity never changes (ignoring air) while the vertical velocity is steadily pulled down by gravity, the combined path traces a parabola. The launch angle decides how the initial speed is divided between horizontal and vertical: a low angle sends the projectile far but flat, a high angle sends it high but short. Switch on air resistance and drag bleeds off speed, collapsing the neat parabola into a shorter, lopsided curve.

Experiments to try in this simulation

1. The 45° maximum: on Earth with air resistance off, sweep the launch angle and watch the range. It peaks at 45°, and angles equally above and below — say 30° and 60° — land in the same place, because complementary angles share a range.

2. Angle pairs: launch at 30°, note the distance, then launch at 60°. Same range, but the 60° shot stays airborne far longer and flies much higher.

3. Gravity matters: fire the identical launch on the Moon, then on Jupiter. Low gravity stretches the range dramatically; high gravity crushes it.

4. Air resistance breaks the symmetry: turn on drag and repeat the 30°/60° test. The complementary-angle rule no longer holds, and maximum range now falls below 45° — closer to how real cannonballs and golf balls actually fly.

Key equations

Neglecting air, the horizontal and vertical positions are x = v₀·cos θ·t and y = v₀·sin θ·t − ½gt². Eliminating time between them gives the parabola. For level ground the time of flight is t = 2·v₀·sin θ / g, and the range is R = v₀²·sin(2θ) / g.

That range formula explains everything the simulator shows: range grows with the square of launch speed, shrinks as gravity g increases, and is largest when sin(2θ) = 1 — that is, at θ = 45°. Air resistance adds a velocity-dependent drag force with no simple closed-form solution, which is exactly why the simulator integrates the motion numerically instead of using a formula.

Real-world applications

Projectile physics governs anything launched and then left to gravity: artillery and ballistics, the arc of a basketball or a long-jumper, water fountains, and the flight of fireworks. Sports scientists use it to find optimal launch angles — the fact that maximum range drops below 45° once drag is included is why a well-struck golf ball or javelin leaves at a lower angle than pure theory suggests. Extended with orbital mechanics, the same two-independent-axes idea is how spacecraft trajectories are first sketched out.