🎯 Projectile Motion

Each of the four physics sub-steps per frame updates velocity from gravity (and drag, if on) and then position from that velocity — semi-implicit Euler integration at dt = 0.005 s. Once the height crosses below zero, the simulator uses the previous and current points to linearly interpolate the exact crossing time and position, which is what the yellow range marker displays; this happens independently of the frame rate, so the reported range and time of flight stay accurate even if your browser drops frames.

Without drag, range R = v₀²sin(2θ)/g is maximised exactly at θ=45° because sin(2θ) peaks at 2θ=90°. Drag force scales with v² (Fₐ = ½ρCₐAv²) and acts for the entire flight, so a steeper 45° shot loses proportionally more energy to drag than a flatter shot with the same vacuum range. The simulator recomputes all five reference arcs and the live trajectory with the exact same drag law, so you can watch the optimal angle visibly drop toward 30°–40° as you increase Cₐ, radius, or speed.

Drag coefficient Cₐ describes the ball's shape (0.47 is a smooth sphere); radius sets the cross-sectional area A=πr² the air pushes against; mass sets how much that push actually decelerates the ball, since acceleration from drag is Fₐ/mass. A light, large-Cₐ, large-radius ball (imagine a beach ball) is dragged down hard and lands short of its vacuum range; a heavy, small, streamlined one (a shot-put) barely deviates from the pure parabola traced in the formula box.

Range, max height, and time of flight all divide by g in the drag-free equations, so every one of them scales as 1/g. Switching from Earth's 9.81 m/s² to the Moon's 1.62 m/s² multiplies range, height, and time of flight by roughly 6×, with everything recomputed instantly — including the five comparison arcs and the envelope curve, which is itself proportional to v₀²/g.

The five fixed-angle arcs (15°, 30°, 45°, 60°, 75°) let you see, without firing repeatedly, how launch angle trades off height for range at the current speed and gravity — notice 15° and 75° land at the same range (they're complementary angles, sin(2·15°)=sin(2·75°)), while 45° reaches farthest without drag. The dashed envelope curve is the ‘parabola of safety’: it traces the outer boundary of every point reachable at the given launch speed from any angle, so anything outside the dashed line simply cannot be hit at that speed, no matter the angle chosen.