### IMPORTS import numpy as np # Linalg import matplotlib.pyplot as plt # Plots

def diff_eqns(state, drag, g=-9.81): """ Evaluates the drag differential for the given state, feed to RK4 'function' parameter Feed to RK4 'function' parameter.""" return np.array([ state[2], # dx state[3], # dy drag(state)*state[2]**2, # dvx g + drag(state)*state[3]**2]) # dvy

def RK4(Nt, dt, state0, t0, function, drag, g): """ 4th order Runga Kutta function for ODE solving """ states = [state0] times = [t0] for n in range(Nt): F1 = function(states[n] , drag, g ) F2 = function(states[n] + 0.5*dt*F1, drag, g ) F3 = function(states[n] + 0.5*dt*F2, drag, g ) F4 = function(states[n] + dt*F3, drag, g ) states.append(np.array(states[n]+(dt/6)*(F1 + 2*F2 + 2*F3 + F4))) times.append(times[n]+dt) # Checks if projectile has hit the ground if states[n+1][1] < 0: return np.array(states), np.array(times) print(f"Maxed out iterations at {Nt} iterations") return np.array(states), np.array(times)

# Task sheet (Table 1): GIVEN CONSTANTS B2m = 4 * 10**(-5) y0 = 10**(4) a = 6.5 * 10**(-3) alpha = 2.5 ### OTHER DEFINED CONSTANTS # mass # Gravitational acceleration, assumed constant g = -9.81 T0 = 293.15 # room temperature in Kelvin # Max number of iterations Nt = (int)(1e30) t0 = 0.0 dt = .1 #Initial Values init_vel = 700 angle = 45 # degrees state0 = np.array([ 0.0, # x 0.0, # y init_vel*np.cos(np.radians(angle)), # vx init_vel*np.sin(np.radians(angle))]) # vy no_drag = lambda state: 0 # Integrates with no drag states, t = RK4(Nt, dt, state0, t0, diff_eqns, no_drag, g)

### PLOTTING fig , ax = plt.subplots () ax.set_title(r'Projectile Motion without Air Resistance, init angle $45^\circ$') ax.set_xlabel('Distance [m]') ax.set_ylabel('Height [m]') # Plot the trajectory plt.plot(states[:,0], states[:,1], label = 'Trajectory') plt.axhline(0, color='black') plt.axvline(0, color='black') ax.legend() plt.show() print(f"{states[-1,0]}m langt")

dts = np.linspace(0.001, 0.1, 100) xs = [] for dt in dts: xs.append(RK4(Nt, dt, state0, t0, diff_eqns, no_drag, g)[0][-1,0])

### PLOTTING fig , ax = plt.subplots () ax.set_title(r'Error as function of timestep') ax.set_xlabel('Time step [s]') ax.set_ylabel('Error [m]') # Plot the trajectory plt.plot(dts, np.array(xs)-49949) plt.axhline(0, color='black') plt.axvline(0, color='black') plt.show()

ez_drag = lambda state: -B2m iso_drag = lambda state: -np.exp(-state[1]/y0) * B2m adia_drag = lambda state: -(1-a*state[1]/T0)**alpha * B2m

# Integrates easy drag dt = 0.01 states_ez, t = RK4(Nt, dt, state0, t0, diff_eqns, ez_drag, g) # Integrates isothermal drag states_iso, t = RK4(Nt, dt, state0, t0, diff_eqns, iso_drag, g) # Integrates adiabatic drag states_adia, t = RK4(Nt, dt, state0, t0, diff_eqns, adia_drag, g)

### PLOTTING fig , ax = plt.subplots () ax.set_title(r'Projectile Motion with Air Resistance, init angle $45^\circ$') ax.set_xlabel('Distance [m]') ax.set_ylabel('Height [m]') # Plot the trajectory plt.plot(states_ez[:,0], states_ez[:,1], label = 'Simple drag') plt.plot(states_iso[:,0], states_iso[:,1], label = 'Isothermal drag') plt.plot(states_adia[:,0], states_adia[:,1], label = 'Adiabatic drag') plt.plot(states[:,0], states[:,1], label = 'No drag') plt.axhline(0, color='black') plt.axvline(0, color='black') ax.legend() plt.show()

### Gets maxima for different air models with different angels n_angles = 1000 angles = np.linspace(1/1000, np.pi/2, n_angles) states0 = np.array([np.zeros(n_angles), # x np.zeros(n_angles), # y init_vel*np.cos(angles), # vx init_vel*np.sin(angles)]) # vy states0 = np.transpose(states0) max_ez = 0 max_iso = 0 max_adia = 0 optimal_ez = 0 optimal_iso = 0 optimal_adia = 0 for n in range(n_angles): ez = RK4(Nt,dt,states0[n],t0,diff_eqns,ez_drag,g)[0] if max(ez[:,0]) > max_ez: max_ez = max(ez[:,0]) optimal_ez = angles[n] iso = RK4(Nt,dt,states0[n],t0,diff_eqns,iso_drag,g)[0] if max(iso[:,0]) > max_iso: max_iso = max(iso[:,0]) optimal_iso = angles[n] adia = RK4(Nt,dt,states0[n],t0,diff_eqns,adia_drag,g)[0] if max(adia[:,0]) > max_adia: max_adia = max(adia[:,0]) optimal_adia = angles[n]

print(f"\t\tDist.(m)\tAngle(deg)") print(f"Easydrag: \t{ max_ez:.0f}m\t\t{np.degrees(optimal_ez):.3f}") print(f"Isodrag: \t{max_iso:.0f}m\t\t{np.degrees(optimal_iso):.3f}") print(f"Adiadrag: \t{max_adia:.0f}m\t\t{np.degrees(optimal_adia):.3f}")

m = 106 #kg v_init = 1640 #m/s state0 = np.array([ 0, 0, v_init*np.cos(optimal_adia), v_init*np.sin(optimal_adia) ]) parisgun_results, t = RK4(Nt,dt,state0,t0,diff_eqns,adia_drag,g)

### PLOTTING fig , ax = plt.subplots () ax.set_title("Paris Gun with adiabatic model") ax.set_xlabel('Distance [m]') ax.set_ylabel('Height [m]') # Plot the trajectory plt.plot(parisgun_results[:,0], parisgun_results[:,1], label = 'Trajectory') plt.axhline(0, color='black') plt.axvline(0, color='black') ax.legend() plt.show() print(f"Paris Gun has a maximum range of {parisgun_results[-1,0]:.0f}m") print(f"Max altitude of {max(parisgun_results[:,1]):.0f}m") print(f"Flight time of {t[-1]:.0f}s")