File:02 Horiz vel.png

#!/usr/bin/env python3
from math import *
import numpy as np
from scipy.integrate import odeint
from scipy.optimize import newton
import matplotlib.pyplot as plt

def projectile_motion(g, mu, xy0, vxy0, tt):
    # use a four-dimensional vector function vec = [x, y, vx, vy]
    def dif(vec, t):
        # time derivative of the whole vector vec
        v = sqrt(vec[2]**2 + vec[3]**2)
        return [vec[2], vec[3], -mu * v * vec[2], -g - mu * v * vec[3]]
    
    # solve the differential equation numerically
    vec = odeint(dif, [xy0[0], xy0[1], vxy0[0], vxy0[1]], tt)
    return vec[:,0], vec[:,1], vec[:,2], vec[:,3] # return x, y, vx, vy

# Parameters of projectile (modelled after a baseball)
g       = 9.81         # Acceleration due to gravity (m/s^2)
rho_air = 1.29         # Air density (kg/m^3)
v0      = 44.7         # Initial velocity (m/s)
alpha0  = radians(75)  # Launch angle (deg.)
m       = 0.145        # Mass of projectile (kg)
cD      = 0.5          # Drag coefficient (spherical projectile)
r       = 0.0366       # Radius of projectile (m)
mu = 0.5 * cD * (pi * r**2) * rho_air / m

# Initial position and launch velocity
x0, y0 = 0.0, 0.0
vx0, vy0 = v0 * cos(alpha0), v0 * sin(alpha0)

T_peak = newton(lambda t: projectile_motion(g, mu, (x0, y0), (vx0, vy0), [0, t])[3][1], 3)
y_max = projectile_motion(g, mu, (x0, y0), (vx0, vy0), [0, T_peak])[1][1]
T = newton(lambda t: projectile_motion(g, mu, (x0, y0), (vx0, vy0), [0, t])[1][1], T_peak + 3)
t = np.linspace(0, T, 501)
x, y, vx, vy = projectile_motion(g, mu, (x0, y0), (vx0, vy0), t)

print("Time of flight: {:.1f} s".format(T))        # returns  6.6 s
print("Horizontal range: {:.1f} m".format(x[-1]))  # returns 43.7 m
print("Maximum height: {:.1f} m".format(y_max))    # returns 53.4 m

# Plot of trajectory
fig, ax = plt.subplots()
(line,) = ax.plot(x, y, "r-", label="Numerical")
ax.set_title(r"Projectile path")
ax.set_aspect("equal")
ax.grid(b=True)
ax.legend()
ax.set_xlabel("$x$ (m)")
ax.set_ylabel("$y$ (m)")
plt.savefig('01 Path.png')

# Plot of velocity components
fig, ax = plt.subplots()
(line,) = ax.plot(t, vx, "b-", label="$v_x$")
ax.set_title(r"Horizontal velocity component")
ax.grid(b=True)
ax.legend()
ax.set_xlabel("$t$ (s)")
ax.set_ylabel("$v_x$ (m/s)")
plt.savefig('02 Horiz vel.png')

fig, ax = plt.subplots()
(line,) = ax.plot(t, vy, "b-", label="$v_y$")
ax.set_title(r"Vertical velocity component")
ax.grid(b=True)
ax.legend()
ax.set_xlabel("$t$ (s)")
ax.set_ylabel("$v_y$ (m/s)")
plt.savefig('03 Vert vel.png')