import numpy as np

import matplotlib.pyplot as plt

# Constants

g = 9.81  # acceleration due to gravity (m/s^2)

h_initial = 0.50  # initial height (m)

# Time array (0 to the time it takes to reach the ground)

t = np.linspace(0, np.sqrt(2 * h_initial / g), 100)

# Calculate height as a function of time (y = h_initial - 1/2 * g * t^2)

h = h_initial - 0.5 * g * t**2

# Plotting the results

plt.figure(figsize=(8, 6))

plt.plot(t, h, label='Height vs Time', color='b')

plt.xlabel('Time (s)')

plt.ylabel('Height (m)')

plt.title('Free Fall: Time vs Height from 0 to 50 cm')

plt.grid(True)

plt.legend()

plt.show()

# Calculate time squared (t^2)

t_squared = t**2

# Plotting the results

plt.figure(figsize=(8, 6))

plt.plot(t_squared, h, label='Height vs Time Squared', color='g')

plt.xlabel('Time Squared (s^2)')

plt.ylabel('Height (m)')

plt.title('Free Fall: Time Squared vs Height from 0 to 50 cm')

plt.grid(True)

plt.legend()

plt.show()

# Constants for air resistance

C_d = 0.47  # drag coefficient for a spherical object

rho = 1.225  # air density in kg/m^3 (at sea level)

A = 0.0014  # cross-sectional area in m^2 (assuming a small ball of ~4.5 cm diameter)

m = 0.005  # mass of the object in kg (5 grams)

# Time array for simulation

t_air = np.linspace(0, 2, 1000)  # longer time array for more realistic simulation with air resistance

dt = t_air[1] - t_air[0]  # time step

# Initialize arrays for velocity and height

v = np.zeros_like(t_air)  # velocity

h_air = np.zeros_like(t_air)  # height

h_air[0] = h_initial  # starting height

# Simulate free fall with air resistance

for i in range(1, len(t_air)):

    F_gravity = m * g

    F_drag = 0.5 * C_d * rho * A * v[i-1]**2

    net_force = F_gravity - F_drag

    a = net_force / m

    v[i] = v[i-1] + a * dt

    h_air[i] = h_air[i-1] - v[i] * dt

    if h_air[i] <= 0:  # stop simulation when object hits the ground

        h_air[i] = 0

        break

# Plotting the results with air resistance

plt.figure(figsize=(8, 6))

plt.plot(t_air, h_air, label='Height vs Time (with air resistance)', color='r')

plt.xlabel('Time (s)')

plt.ylabel('Height (m)')

plt.title('Free Fall with Air Resistance: Time vs Height from 0 to 50 cm')

plt.grid(True)

plt.legend()

plt.show()

The terminal velocity of an object in free fall is the constant speed it reaches when the force of air resistance (drag) equals the gravitational force, and there is no further acceleration. At this point, the object falls at a constant speed.

The terminal velocity can be calculated using the following formula:

Where:

• vt is the terminal velocity,

• m is the mass of the object,

• g is the acceleration due to gravity (9.81 m/s²),

• ρ is the air density (about 1.225 kg/m³ at sea level),

• Cd​ is the drag coefficient (depends on the shape of the object, e.g., 0.47 for a sphere),

• A is the cross-sectional area of the object.

Steps for Calculating Terminal Velocity:

1. Determine the parameters:

   • Mass of the object (m).

   • Cross-sectional area (A) can be calculated for a sphere using the formula A=πr², where r is the sphere's radius.

   • Drag coefficient (Cd​) depends on the shape. For a sphere, Cd​ is typically around 0.47.

   • Air density (ρ) is about 1.225 kg/m³ at sea level.

2. Use the terminal velocity formula: Plug these values into the formula to calculate the terminal velocity.

Let me calculate the terminal velocity for the object we used in the previous simulation (5 g mass, 4.5 cm diameter sphere):

The calculated terminal velocity for the 5g spherical object (with a 4.5 cm diameter) is approximately 10.35 m/s. This means that once the object reaches this speed during free fall, the force of air resistance will balance the gravitational force, and the object will fall at this constant velocity.

# Parameters for the terminal velocity calculation

r = 0.045 / 2  # radius of the sphere in meters (4.5 cm diameter)

A_sphere = np.pi * r**2  # cross-sectional area of the sphere

# Calculate terminal velocity

v_terminal = np.sqrt((2 * m * g) / (rho * C_d * A_sphere))

v_terminal