Projectile motion
Launch a projectile and watch physics unfold. Angle, velocity, gravity, air resistance — adjust them all and see how drag transforms the clean parabola into something messier and more real.
x = v0cosθ · t y = v0sinθ · t − ½gt² Fdrag = −½CdρAv²
The ideal case
Without air resistance, projectile motion is governed entirely by gravity acting vertically while horizontal velocity stays constant. The trajectory is a perfect parabola. The range formula R = v²sin(2θ)/g predicts maximum range at exactly 45°, and the motion is completely time-reversible — the projectile follows the same path going up as coming down.
Adding air resistance
Real projectiles encounter drag, a force proportional to the square of velocity (for high-Reynolds-number flow). The drag force F = ½CdρAv² acts opposite to the velocity vector. This breaks the symmetry of the parabola: the descending arc steepens, the range shortens, and the optimal launch angle shifts below 45° — typically to around 38–43° depending on drag strength.
Why the optimal angle shifts
With drag, higher angles mean longer time in the air, which means more time for drag to slow the projectile. A lower angle gets the projectile to the ground faster with less total drag work. The exact optimal angle depends on the ratio of gravitational force to drag force — a dimensionless parameter that characterizes the regime.
Newton’s laws at work
Projectile motion is a direct application of Newton’s second law: F = ma. The only force in the ideal case is gravity (constant, downward). With drag, the equations become nonlinear — the drag depends on velocity, which depends on the drag. No closed-form solution exists for quadratic drag; you must integrate numerically, which is exactly what this simulation does with a fourth-order Runge-Kutta method.