line of fire

drop height

initial position

D = (7.914071754582752, 4.475058574160522)
E = (29.052409624748485, 14.495970907423413)
d: x(E) - x(D)
h: y(E) - y(D)
a: y = y(D)
b: x = x(E)
F = (12.593635679206814, 6.670853954176416)
u = Vector(D, F)
c = Ray(D, F)
G = Intersect(a, b)
α = Angle(G, D, F)
I = Point(b)
K = Point(a)
H: (x(D), y(I))
J: (x(K), y(E))
distanceHI = Distance(H, I)
distanceKJ = Distance(K, J)
distanceDF = Distance(D, F)
v: (5 * abs(u))
text1: "$ v=" + (LaTeX[v]) + " $"
e = Segment(H, I)
duration: d / v / cos(α)
t = 0.03 (slider from 0 to 1.4, step 0.01)
g = 9.81
P: (x(D) + ((v * cos(α)) * t), y(D) + ((v * sin(α)) * t) - (((0.5 * g) * t) * t))
M: (x(E), y(E) - (((0.5 * g) * t) * t))
f = OrthogonalLine(P, a)
L = Intersect(f, c)
i = Segment(P, L)
j = Segment(M, E)
k = Line(P, M)
l = false
m = false
TextHI: " " + (LaTeX[distanceHI]) + " m"
TextKJ: " " + (LaTeX[distanceKJ]) + "m"
n = Segment(J, K)
xshift = 22.4 (slider from 0 to 30, step 0.1)
yshift = 3.1 (slider from -5 to 5, step 0.1)
A: (3 + xshift, 11 + yshift - i)
B: (8 + xshift, 8 + yshift - i)
C: (6 + xshift, 14.81624 + yshift - i)
N = (21.445606297776486, 19.926670703702865)
O = (21.445606297776486, 15.926670703702865)
Q = (35.44560629777658, 15.926670703702865)
R = (27.831292647967953, 17.432847228194664)
S = (29.715148518838994, 17.393993502790906)
T = (27.950496924272727, 18.177873955099486)
o = false
p = Segment(P, M)
Z: (x(M), y(M))
q = Circle(Z, 4)
U = Point(q)
V = Point(q)
W = Point(q)
dead = 0
elevation: atan(h / d)
accur: abs(elevation - α)
A_1: (x(M), y(M))
B_1: (x(M) + 3.5, y(M) + 1)
r = Circle(A_1, B_1)
s = Line(B_1, a)
C_1 = Intersect(r, s, 2)
a_1 = OrthogonalLine(C_1, s)
D_1 = Intersect(r, a_1, 2)
w: x(P) > x(E)
z: accur  <  0.005
b_1: (w * z)
c_1 = Circle(P, 5.79)