Thank you for uploading the problem. I will work on sketching and solving the key parts of the given tasks. I’ll start by solving or visualizing the first question (M4.2) and providing diagrams or graphs where needed. Let me extract the main details and proceed.import numpy as np import matplotlib.pyplot as plt

Constants for M4.2

m = 10.0 # mass in kg k = 1.96e3 # spring constant in N/m z1 = 0.6 # initial height in m z3 = -0.1 # compressed spring position in m z4 = 0.4 # height to fall from in m g = 9.81 # gravity in m/s^2

Potential energy of the system as a function of z

z_range = np.linspace(-0.3, 0.6, 500) potential_energy = m * g * z_range + 0.5 * k * np.maximum(0, -z_range)**2

Plotting the total potential energy

plt.figure(figsize=(8, 6)) plt.plot(z_range, potential_energy, label="Total Potential Energy") plt.axvline(x=z3, color='r', linestyle='--', label="Spring Compression (z3)") plt.axvline(x=0, color='g', linestyle='--', label="Spring Rest Position (z=0)") plt.title("Total Potential Energy of the System") plt.xlabel("z-position (m)") plt.ylabel("Potential Energy (J)") plt.legend() plt.grid() plt.show() <<ImageDisplayed>>