Physics · Simulator 001
Launch a ball and compare its path with earlier launches. Change the angle, speed, launch height, gravity and air drag, then see how range, peak height and flight time respond.
Without air resistance the only force is gravity, pointing straight down. Horizontal velocity stays constant and vertical velocity changes at a steady rate g:
x(t) = v·cos θ · t
y(t) = h + v·sin θ · t − ½ g t²
From level ground (h = 0) the range is R = v² sin 2θ / g, which is largest at θ = 45°. When you launch from a height, the projectile stays in the air longer, so a slightly lower angle wins.
With drag switched on, the simulator adds a force opposite to the velocity, proportional to speed squared: a = −(k/m)|v|v. You set k/m directly in 1/m. There is no simple formula for the range, so the path is integrated numerically with small time steps (RK4, 1 ms).
This is an educational model. Real balls feel lift from spin, drag that varies with speed, and wind.
A ball is thrown at 15 m/s from a 10 m cliff with no drag. Is 45° still the angle that gives the longest range? Predict first, then test 45°, 40° and 35°.
No. About 36.2° gives the longest range: about 31.4 m, against 30.5 m at 45°. Starting high adds flight time, so it pays to trade some vertical speed for more horizontal speed. The exact optimum is sin θ = v / √(2v² + 2gh). As h grows, this tends towards 0°; at h = 0 it gives exactly 45°.