Basketball Problem

Family

MinVelocity

Envelope

α = 1.1780972450961724 rad (slider from 0*° to 90*°, step 0.1*°)
U = 7.2 (slider from 0 to 20, step 0.1)
H = 2 (slider from -20 to 20, step 0.1)
R = 4.4 (slider from 0 to 20, step 0.1)
k = 4 (slider from -5 to 6, step 0.1)
e_1(x) = H + U^(2) / (2 * 9.8) - 9.8 / (2 * U^(2)) * x^(2)
Hoop: (R, k)
f_1(x, α) = x * tan(α) - 9.8 * x^(2) / (2 * U^(2)) * (1 + tan(α)^(2)) + H
β = 0.10471975511965977 rad (slider from 1*° to 6*°, step 1*°)
g_1(x, b) = x * tan(b) - 9.8 * x^(2) / (2 * U^(2)) * (1 + tan(b)^(2)) + H
c: y = f_1(x, α)
a = false
H_L: (x(Hoop) - 0.225, y(Hoop))
H_R: (x(Hoop) + 0.225, y(Hoop))
b = Segment(H_L, H_R)
Board: (x(Hoop) + 0.5, y(Hoop))
B_T: (x(Board), y(Board) + 1)
B_B: (x(Board), y(Board) - 0.3)
d = Segment(Hoop, B_B)
e = Segment(B_B, B_T)
B_S: (x(Board) + 0.75, y(Board))
B_0: (x(Board) + 1.15, 0)
f = Segment(B_S, B_0)
g = Segment(B_S, Board)
f_2(x, α) = x * tan(α) - x^(2) * (H + R * tan(α) - k) / R^(2) + H
h: y = f_2(x, α)
i = false
e_2(x) = H + (R^(2) - (H - k) * sqrt((H - k)^(2) + R^(2)) + (H - k)^(2)) / (2 * sqrt((H - k)^(2) + R^(2))) - sqrt((H - k)^(2) + R^(2)) * x^(2) / (2 * (R^(2) - (H - k) * sqrt((H - k)^(2) + R^(2)) + (H - k)^(2)))
A = Point(yAxis)
B = Point(yAxis)
C = (-8.81712702786856, -7.5395691240477944)
D = (-8.955521894827529, 8.907279705238308)
poly1, a_1, b_1, c_1, d_1 = Polygon(A, B, C, D)
E = Intersect(a_1, xAxis)
F = Point(xAxis)
G = (19.5, -7.5)
poly2, e_3, f_3, g_2, b_2 = Polygon(E, F, G, B)
list = Sequence(y = g_1(x, n), n, 0*°, 360*°, β)
l = false
P: (α * 180 / π, sqrt(9.81 * R^(2) * (1 + tan(α)^(2)) / (2 * (R * tan(α) + H - k))))
loc1 = Locus(P, α)
Q: (α * 180 / π, U)