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The best kick with air drag (a golden-section search)

Script by Ordinate 2D ✓ · Mechanics

A football kicked at 25 m/s in air: the script searches for the angle of the longest kick, which is not 45°.

"""The best launch angle with air drag: a golden-section search for the longest kick.

In a vacuum a ball flies furthest at 45°. With air drag it does not: the script kicks a football (0.43 kg, 11 cm radius)
at 25 m/s, measures how far it lands, and narrows down the best angle with a golden-section search, each kick telling it
which part of the interval to drop. Every kick is a row in the Script sheet; the last line says where the best angle is.

Runs in the studio (Script tab → Run) and locally:  python3 optimal_angle_drag.py
"""
import math
import studio as s

V0 = 25.0              # m/s
LO, HI = 20.0, 60.0    # search interval, degrees
TOL = 0.5              # stop when the interval is this narrow, degrees
PHI = (math.sqrt(5) - 1) / 2

s.clear()
s.gravity = (0, -9.81)
s.air_density = 1.225
s.ground(-0.2, width=200)
ball = s.circle(0, 0, 0.11, mass=0.43, name="Football", color="#e5484d", collide=False)


def search():
    """A golden-section search for the maximum: yields an angle, is sent back its range."""
    a, b = LO, HI
    c, d = b - PHI * (b - a), a + PHI * (b - a)
    fc, fd = (yield c), (yield d)
    while b - a > TOL:
        if fc > fd:
            b, d, fd = d, c, fc
            c = b - PHI * (b - a)
            fc = yield c
        else:
            a, c, fc = c, d, fd
            d = a + PHI * (b - a)
            fd = yield d
    print(f"best angle ≈ {(a + b) / 2:.1f}° (in a vacuum 45°; vacuum range {V0 ** 2 / 9.81:.1f} m)")


searcher = search()
angle = next(searcher)
kicks = 0


def kick(deg):
    a = math.radians(deg)
    ball.position = (0, 0)
    ball.velocity = (V0 * math.cos(a), V0 * math.sin(a))


kick(angle)


@s.on_step
def watch(t, dt):
    global angle, kicks
    if ball.vy < 0 and ball.y <= 0:                  # back down at the launch height
        rng = ball.x - ball.y * ball.vx / ball.vy
        kicks += 1
        s.record(kick=kicks, angle=angle, range=rng)
        print(f"kick {kicks}: {angle:5.2f}° → {rng:6.2f} m")
        try:
            angle = searcher.send(rng)
        except StopIteration:
            s.stop()
            return
        kick(angle)


s.run(120)