Newton's Laws in Python
Newton's three laws of motion form the foundation for understanding all classical mechanics. These laws describe how objects behave in response to forces, and they can be expressed mathematically to predict and analyze motion.
Newton's First Law (Law of Inertia):
An object at rest remains at rest, and an object in motion continues in motion with constant velocity unless acted upon by a net external force.
Newton's Second Law:
The acceleration a of an object is directly proportional to the net force F acting on it and inversely proportional to its mass m. This law is represented mathematically as:
F = m * a
or, rearranged,
a = F / m
Newton's Third Law:
For every action, there is an equal and opposite reaction. If object A exerts a force on object B, then object B exerts an equal force in the opposite direction on object A.
1234567891011121314151617181920212223242526272829303132333435# Simulating motion under a constant force using Newton's Second Law import numpy as np import matplotlib.pyplot as plt # Parameters mass = 2.0 # kg force = 10.0 # N time_total = 5.0 # seconds dt = 0.1 # time step in seconds # Initial conditions velocity = 0.0 # m/s position = 0.0 # m # Lists to store results time_points = np.arange(0, time_total + dt, dt) positions = [] velocities = [] for t in time_points: acceleration = force / mass # Newton's second law velocity += acceleration * dt position += velocity * dt velocities.append(velocity) positions.append(position) # Plotting position vs. time plt.plot(time_points, positions, label="Position (m)") plt.plot(time_points, velocities, label="Velocity (m/s)") plt.xlabel("Time (s)") plt.ylabel("Value") plt.title("Motion Under Constant Force") plt.legend() plt.show()
In the simulation above, you apply a constant force to an object and use Newton's second law (F = m * a) to calculate the acceleration. At each time step, the code updates the velocity by adding the product of acceleration and the time interval (dt). The position is then updated by adding the new velocity multiplied by dt. By repeating this process, you can track how the object speeds up and moves farther as time passes under a constant force.
123456789101112131415161718192021222324252627282930313233343536373839404142# Simulating motion with changing forces (friction and thrust) import numpy as np import matplotlib.pyplot as plt # Parameters mass = 2.0 # kg thrust = 12.0 # N (applied force) friction_coefficient = 0.8 # N (constant friction) time_total = 8.0 # seconds dt = 0.1 # time step in seconds # Initial conditions velocity = 0.0 # m/s position = 0.0 # m # Lists to store results time_points = np.arange(0, time_total + dt, dt) positions = [] velocities = [] for t in time_points: # Friction opposes motion, so subtract it from thrust if moving forward friction = friction_coefficient if velocity > 0 else 0 net_force = thrust - friction acceleration = net_force / mass velocity += acceleration * dt position += velocity * dt velocities.append(velocity) positions.append(position) # After 4 seconds, turn off the thrust if t >= 4.0: thrust = 0 # Plotting position and velocity plt.plot(time_points, positions, label="Position (m)") plt.plot(time_points, velocities, label="Velocity (m/s)") plt.xlabel("Time (s)") plt.ylabel("Value") plt.title("Motion with Thrust and Friction") plt.legend() plt.show()
1. Which of Newton's laws explains why a stationary object remains at rest unless acted upon?
2. How does increasing mass affect acceleration for a given force in the simulation?
3. Fill in the blank: According to Newton's second law, acceleration is equal to force divided by ___?
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Newton's three laws of motion form the foundation for understanding all classical mechanics. These laws describe how objects behave in response to forces, and they can be expressed mathematically to predict and analyze motion.
Newton's First Law (Law of Inertia):
An object at rest remains at rest, and an object in motion continues in motion with constant velocity unless acted upon by a net external force.
Newton's Second Law:
The acceleration a of an object is directly proportional to the net force F acting on it and inversely proportional to its mass m. This law is represented mathematically as:
F = m * a
or, rearranged,
a = F / m
Newton's Third Law:
For every action, there is an equal and opposite reaction. If object A exerts a force on object B, then object B exerts an equal force in the opposite direction on object A.
1234567891011121314151617181920212223242526272829303132333435# Simulating motion under a constant force using Newton's Second Law import numpy as np import matplotlib.pyplot as plt # Parameters mass = 2.0 # kg force = 10.0 # N time_total = 5.0 # seconds dt = 0.1 # time step in seconds # Initial conditions velocity = 0.0 # m/s position = 0.0 # m # Lists to store results time_points = np.arange(0, time_total + dt, dt) positions = [] velocities = [] for t in time_points: acceleration = force / mass # Newton's second law velocity += acceleration * dt position += velocity * dt velocities.append(velocity) positions.append(position) # Plotting position vs. time plt.plot(time_points, positions, label="Position (m)") plt.plot(time_points, velocities, label="Velocity (m/s)") plt.xlabel("Time (s)") plt.ylabel("Value") plt.title("Motion Under Constant Force") plt.legend() plt.show()
In the simulation above, you apply a constant force to an object and use Newton's second law (F = m * a) to calculate the acceleration. At each time step, the code updates the velocity by adding the product of acceleration and the time interval (dt). The position is then updated by adding the new velocity multiplied by dt. By repeating this process, you can track how the object speeds up and moves farther as time passes under a constant force.
123456789101112131415161718192021222324252627282930313233343536373839404142# Simulating motion with changing forces (friction and thrust) import numpy as np import matplotlib.pyplot as plt # Parameters mass = 2.0 # kg thrust = 12.0 # N (applied force) friction_coefficient = 0.8 # N (constant friction) time_total = 8.0 # seconds dt = 0.1 # time step in seconds # Initial conditions velocity = 0.0 # m/s position = 0.0 # m # Lists to store results time_points = np.arange(0, time_total + dt, dt) positions = [] velocities = [] for t in time_points: # Friction opposes motion, so subtract it from thrust if moving forward friction = friction_coefficient if velocity > 0 else 0 net_force = thrust - friction acceleration = net_force / mass velocity += acceleration * dt position += velocity * dt velocities.append(velocity) positions.append(position) # After 4 seconds, turn off the thrust if t >= 4.0: thrust = 0 # Plotting position and velocity plt.plot(time_points, positions, label="Position (m)") plt.plot(time_points, velocities, label="Velocity (m/s)") plt.xlabel("Time (s)") plt.ylabel("Value") plt.title("Motion with Thrust and Friction") plt.legend() plt.show()
1. Which of Newton's laws explains why a stationary object remains at rest unless acted upon?
2. How does increasing mass affect acceleration for a given force in the simulation?
3. Fill in the blank: According to Newton's second law, acceleration is equal to force divided by ___?
Thanks for your feedback!
