%matplotlib inline import numpy as np import matplotlib.pyplot as pl

# # Below is the raw data for the position as a function of time for the muffin cup. The "split" function was used to # separate the data into a list of strings. It then will print out a statement explaining the number of data points. # slist=[] tlist=[] data = """0.000000000E0 -2.688162330E0 3.336670003E-2 -4.301059729E0 6.673340007E-2 -5.376324661E0 1.001001001E-1 -6.989222059E0 1.334668001E-1 -1.129028179E1 1.668335002E-1 -1.451607658E1 2.002002002E-1 -2.043003371E1 2.335669002E-1 -2.526872591E1 2.669336003E-1 -3.118268303E1 3.003003003E-1 -3.870953756E1 3.336670003E-1 -4.623639208E1 3.670337004E-1 -5.430087907E1 4.004004004E-1 -6.236536606E1 4.337671004E-1 -7.150511799E1 4.671338005E-1 -8.010723744E1 5.005005005E-1 -8.924698937E1 5.338672005E-1 -9.892437376E1 5.672339006E-1 -1.080641257E2 6.006006006E-1 -1.177415101E2 6.339673006E-1 -1.274188945E2 6.673340007E-1 -1.370962788E2 7.007007007E-1 -1.467736632E2 7.340674007E-1 -1.575263126E2 7.674341008E-1 -1.672036969E2 8.008008008E-1 -1.768810813E2 8.341675008E-1 -1.865584657E2 8.675342009E-1 -1.973111150E2 9.009009009E-1 -2.075261319E2 9.342676009E-1 -2.182787812E2 9.676343010E-1 -2.284937981E2 """.splitlines() # split this string on the "newline" character. print("We have", len(data), "data points.")

# # This code will take the lists created above and break them into real numbers with reasonable units. # tlist = [] ylist = [] for s in data: t,y = s.split() # break string in two t=float(t) # convert time to float y=float(y)/100.0 # convert distanct (in meters) to float tlist.append(t) ylist.append(y) print ("tlist=",tlist) print ("ylist=",ylist)

# # This will plot the raw data above. The x-axis will use the tlist and the y-axis will use the ylist. # pl.plot(tlist, ylist) pl.title("raw data") pl.xlabel("time(s)") pl.ylabel("y(m)") pl.grid()

vlist = [] # Velocity list (computed velocities from experimental data) tvlist = [] # time list (times for corresponding velocities) for i in range(1,len(tlist)): dy=ylist[i]-ylist[i-1] # This calculates the change in the ylist. dt=tlist[i]-tlist[i-1] # This calculates the change in the tlist. vlist.append(dy/dt) # This addes all the dy/dt values to the vlist tvlist.append((tlist[i]+tlist[i-1])/2.0) # This plots the new values of (tv,v) pl.plot(tvlist,vlist,'g.') pl.title("Velocity graph") pl.xlabel("time(s)") pl.ylabel("$v_y$ (m/s)") pl.grid()

m=0.0035 # kg of the cupcake g=9.8 # m/s of gravity b=0.00375 # drag coefficient v=0.0 # initial velocity dt = (tlist[-1]-tlist[0])/(len(tlist)-1) # time per frame in original video t=0.0 vclist = [v] tclist = [t] def deriv(v, t): return b*v**2/m - g # dv/dt equation for i in range(len(tlist)): #Creating the Euler's Method graph dv = deriv(v,t)*dt v += dv t += dt vclist.append(v) tclist.append(t) pl.title("Comparison of experimental and drag model") pl.xlabel("time(s)") pl.ylabel("velocity (m/s)") pl.plot(tclist, vclist, 'r-',tvlist,vlist,'g.') #comparing the Euler's method to the raw data

# # Analytical solution vs Euler Model # v=0.0 # initial velocity Euler v2=0.0 # initial velocity Analytical dt = (tlist[-1]-tlist[0])/(len(tlist)-1) # time per frame in original video t=0.0 t2=0.0 vclist = [v] tclist = [t] vc2list = [v2] #Analytical vclist tc2list = [t2] #Analytical tclist def deriv(v, t): return b*v**2/m - g # dv/dt equation def funct(v,t): #I derived this equation by taking the integral of the dv/dt equation. Since everything is a constant return b*v**2*t/m - g*t # the integral was easy. All I had to do was add a t to the components. for i in range(len(tlist)): #Creating the Euler's Method graph dv = deriv(v,t)*dt vt = funct(v2,t2) v += dv t += dt vclist.append(v) tclist.append(t) vc2list.append(v) tc2list.append(t) fig, axs = pl.subplots(2) # For this graph setup I used the referenced this website for the code: axs[0].set_title('Euler Model') # https://matplotlib.org/devdocs/gallery/subplots_axes_and_figures/subplots_demo.html axs[1].set_title('Analytical Solution') # Since the graphs were so similar, the graphs would be overlapping each other axs[0].plot(tclist, vclist,'r-') # Because of this, I compared them on top of each other axs[1].plot(tc2list,vc2list, 'g-') for ax in axs.flat: ax.set(xlabel='time(s)', ylabel='velocity(m/s)') for ax in axs.flat: ax.label_outer()