create python code to figure out start velocity

Question

Have a task and need to figure out the start velocity of an iron ball shot vertically upwards. We know this information.

Answer 1

The values from your image are as follows:

• mass sphere: 263.8602726 kg
• drag coefficient: 0.47
• air density: 1.225 kg/m³
• max height: 80 m
• acceleration due to gravity (G): 9.805 m/s²

---

Assuming you can neglect air resistance, you could use the kinematic equation:

v^2 + u^2 = 2as

where:

• v is the final velocity (0 m/s at the maximum height).
• u is the start velocity.
• a is the acceleration (in this case, -9.805 m/s² due to gravity).
• s is the displacement (80 m).

We can rearrange the equation to solve for the start velocity u:

u = sqrt(v^2 -2as)

Implementing this in Python is relatively straightforward:

def calculate_start_velocity(displacement: float,
                             acceleration: float) -> float:
  """Calculate the start velocity for an object given its displacement and acceleration.

  Args:
      displacement: The displacement of the object (maximum height reached).
      acceleration: The acceleration (should be negative for gravity).

  Returns:
      float: The calculated start velocity.
  """
  final_velocity_squared = 0  # Since final velocity at max height is 0.
  start_velocity_squared = final_velocity_squared - 2 * acceleration * displacement
  start_velocity = start_velocity_squared**0.5
  return start_velocity


def main() -> None:
  """Main function to calculate and print the start velocity."""
  max_height = 80.0  # Displacement.
  gravity = -9.805  # Acceleration due to gravity.
  start_velocity = calculate_start_velocity(max_height, gravity)
  print(f'Initial velocity of the sphere: {start_velocity:.2f} m/s')


if __name__ == '__main__':
  main()

Output:

Initial velocity of the sphere: 39.61 m/s

---

Assuming that air resistance is supposed to be taken into account you can use scipy.optimze.newton (which uses the secant method when the derivative of the function is not provided):

import numpy as np
from scipy.optimize import newton


def find_initial_velocity(v0, mass, drag_coefficient, air_density, diameter, g,
                          max_height):
  """
  Compute the difference between the calculated max height and actual 
  max height for a given initial velocity considering air resistance.
  """
  # Time step for numerical simulation
  dt = 0.01  # in seconds
  velocity = v0
  height = 0
  while velocity > 0:
    # Drag force
    area = np.pi * (diameter / 2)**2
    drag_force = 0.5 * drag_coefficient * air_density * area * velocity**2
    # Net acceleration (both gravity and drag slow down the sphere)
    acceleration = g + drag_force / mass
    # Update velocity and height
    velocity -= acceleration * dt
    height += velocity * dt
  return height - max_height


def main():
  """Calculate the initial velocity of a sphere shot upwards."""
  # Given constants
  mass = 263.8602726  # in kg
  density_of_iron = 7874  # in kg/m^3
  drag_coefficient = 0.47
  air_density = 1.225  # in kg/m^3
  max_height = 80  # in m
  g = 9.805  # in m/s^2
  # Calculate diameter of the sphere
  volume = mass / density_of_iron
  radius = (3 * volume / (4 * np.pi))**(1 / 3)
  diameter = 2 * radius
  # Initial guess for the initial velocity
  initial_guess = np.sqrt(2 * g * max_height)
  # Find the initial velocity using the Secant method
  initial_velocity = newton(find_initial_velocity,
                            initial_guess,
                            args=(mass, drag_coefficient, air_density,
                                  diameter, g, max_height))

  print(f'Initial velocity of the sphere: {initial_velocity:.2f} m/s')


if __name__ == '__main__':
  main()

Output:

Initial velocity of the sphere: 39.88 m/s