Challenge: Solving the Optimisation Problem

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Let's consider a physics-related optimization problem where we need to find the maximum height reached by an object thrown vertically upward with a given initial velocity.

We have the following equation:
h = v * t - 0.5 * g * t**2
that describes the motion of an object.

Our task is to find the time t when the object reaches its maximum height and then find the maximum height h_max.

Calculate the derivatives of the first and second order for the h function.
Find critical points of h function.
Check if these critical points are points of the maximum of the function h.

Løsning

import sympy as sp

# Define the symbols

t = sp.symbols('t')

# Define the height function h(t)

v = 15.0 # Initial velocity in m/s

g = 9.81 # Acceleration due to gravity in m/s^2

h = v * t - (1/2) * g * t**2

# Calculate the first and second derivatives of h(t)

h_prime = sp.diff(h, t)

h_double_prime = sp.diff(h_prime, t)

# Find the critical points by solving h'(t) = 0

critical_points = sp.solve(h_prime, t)

# Identify the maximum height by evaluating h''(t) at each critical point

maximum_height = -1

for t_critical in critical_points:

h_double_prime_value = h_double_prime.subs({t:t_critical})

if h_double_prime_value < 0:

maximum_height = h.subs({t:t_critical})

# Print the result

if maximum_height != -1:

print(f'The object reaches its maximum height of {maximum_height:.2f} meters at time t = {t_critical:.2f} seconds.')

else:

print('There is no maximum height (object doesn\'t reach the ground).')

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Sektion 3. Kapitel 5