#224 Shooting method to solve system of ODEs using python

Let y(t) be the altitude of the missile at time t. The gravity g = 9.8 m/s^2. We want to have the missile at 50 m off the ground after t = 5 s after launch.
Find the velocity at launch v0=y'(0)? Ignore the drag of the air resistance.
Boundary conditions are y(0) = 0 and y(5) = 50.
d^2y/dt^2=-g
or
dy/dt=v
dv/dt=-g

The result should be y'(0)=34.5 m/s.

import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import solve_ivp
from scipy.optimize import fsolve
import warnings
warnings.filterwarnings("ignore", category=np.VisibleDeprecationWarning) 
plt.style.use("seaborn-poster")
F = lambda t, s: np.dot(np.array([[0,1],[0,-9.8/s[1]]]),s)
t_span = np.linspace(0, 5, 100)
y0 = 0
t_eval = np.linspace(0, 5, 100)

def objective(v0):
    sol = solve_ivp(F, [0, 5], [y0, v0], t_eval = t_eval)
    y = sol.y[0]
    return y[-1] - 50

v0, = fsolve(objective, 10)
print("v0 =",v0)
sol = solve_ivp(F, [0, 5], [y0, v0], t_eval = t_eval)
plt.figure(figsize = (10, 8))
plt.plot(sol.t, sol.y[0])
plt.plot(5, 50, "ro")
plt.xlabel("time (s)")
plt.ylabel("altitude (m)")
plt.title(f"root finding v={v0} m/s")
plt.savefig('ex224.png', dpi=72)
plt.show()