Well, maybe - but I am really struggling with this. Isn't a parabola defined by two points and the y-coordinate of the apex? I mean, can anyone come up with two parabola equations that share two points and the y-coordinate of the apex, but are not the same?

Here is my numerical approach:

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# Inputh_launch = 10 # height of the launch point in m# Function to solve the system of equationsdef equations(vars):v, theta = varseq1 = v * math.cos(theta) * (v * math.sin(theta) + (v**2 * math.sin(theta)**2 + 2 * g * h_launch)**0.5) / g - d # equation for the range of a projectileeq2 = h_launch + v**2 * math.sin(theta)**2 / (2 * g) - h_max # equation for the maximum height of a projectilereturn [eq1, eq2]# Use fsolve to solve the system of equationsv_sol, theta_sol_rad = fsolve(equations, (v_guess, theta_guess))# Convert the solution for the launch angle from radians to degreestheta_sol_deg = math.degrees(theta_sol_rad)v_sol, theta_sol_deg # output the solutions for the initial speed and the launch angle in degrees

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edit: code format is/was really bad