Mechanics Without Newton: Empirical Confirmation Of Motion Laws

Newton's laws of motion are important because they are the foundation of classical mechanics, a main branch of physics. Newton's second law of motion explains how force can change the acceleration of an object and how the acceleration and mass of the same object are related. The second law is more quantitative and is used extensively to calculate what happens in situations involving a force. The law is sometimes presented as a definition of force, i.e., a force is that which exists when an inertial observer sees a body accelerate. Newton's second law can be applied to daily life to a great extent. For example, in Formula One racing, engineers try to keep the mass of cars as low as possible. But can a mechanics be empirically confirmed without Newton's second law?

The empirical laws of Kepler

• The orbit of a planet is an ellipse with the Sun at one of the two foci.
• A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
• The square of a planet's orbital period is proportional to the cube of the length of the semi-major axis of its orbit.

Kepler's laws replaced circular orbits and epicycles in the heliocentric theory of Nicolaus Copernicus with elliptical orbits and explained how planetary velocities vary. Kepler's laws imply that the speed of revolution of a planet around the sun is not uniform but changes throughout the planet's year. This speed is fastest when the planet is nearest the sun (perihelion) and slowest when the planet is farthest away (aphelion).

While Kepler's laws are empirical, they do not require Newton's second law to be empirically confirmed. This is because Kepler's laws are based on observations and measurements, and they describe the motion of planets without relying on Newton's theories of motion or gravity.

The law of inertia

Newton's first law, also known as the law of inertia, states that an object at rest or in motion will remain at rest or in motion unless acted upon by an unbalanced force. This tendency to resist changes in a state of motion is inertia. Inertia is a property of matter in which an object or mass resists any change in its velocity, or speed and direction. In other words, matter will tend to keep moving in the same direction at a constant speed when there are no forces or other matter to influence change.

Newton's second law defines a force to be equal to change in momentum (mass times velocity) per change in time. This is also known as the equation of force: F=MA. Newton's second law can help us determine the new values of velocity and mass if we know how big the force is. For example, if we know the difference between the conditions at point “1” and the conditions at point “0”, we can calculate the force using the following equation:

> \(\LARGE F = \frac{m \cdot (V_1 – V_0)}{t_1 – t_0}\)

Newton's second law also tells us that the amount of acceleration is proportional to the size of the force. The amount of acceleration is also inversely proportional to the mass of the object; for equal forces, a heavier object will experience less acceleration than a lighter object.

The relationship between force and acceleration

Newton's three laws of motion describe the relationship between the forces acting on a body and the motion of the body. Newton's second law, in particular, states that the force acting on an object is equal to the product of its mass and its acceleration, or F = ma. This means that the amount of acceleration is directly proportional to the force applied and inversely proportional to the mass of the object. For example, a heavier object will experience less acceleration than a lighter object when subjected to the same external force.

Newton's second law can be used to determine the new values of velocity and mass if the force is known. This is calculated using the formula:

> \(\LARGE F = \frac{m_1 \cdot V_1 – m_0 \cdot V_0}{t_1 – t_0} \)

This formula assumes that the mass remains constant, which is a good assumption for an airplane, for example, as the only change in mass would be due to fuel consumption. However, for objects with changing mass, such as a bottle rocket, we can only look at changes in momentum. In such cases, Newton's second law can be simplified to:

> \(\LARGE F = \frac{m \cdot (V_1 – V_0)}{t_1 – t_0} \)

This equation demonstrates the relationship between force and acceleration, where the change in velocity divided by the change in time gives us the acceleration.

Newton's laws of motion, including the second law, are foundational to classical mechanics. They provide a mathematical framework for understanding the behaviour of objects in motion and at rest, and their interactions with external forces. By combining his knowledge of celestial motions with the study of events on Earth, Newton showed that a single theory of mechanics could explain both.

The behaviour of objects with unbalanced forces

Newton's laws of motion explain the relationship between a physical object and the forces acting upon it. Newton's first law states that every object will remain at rest or in uniform motion in a straight line unless compelled to change its state by the action of an external force. This tendency to resist changes in a state of motion is inertia.

When the net force on a body is equal to zero, Newton's second law states that the body does not accelerate, and it is said to be in mechanical equilibrium. The second law also defines a force to be equal to the change in momentum (mass times velocity) per change in time.

Unbalanced forces refer to forces acting on an object that are not canceled out by opposing forces, resulting in a change in the object's motion. When an unbalanced force is applied to an object, it will either start moving if at rest, speed up, slow down, or change direction if already in motion. This concept is fundamental to understanding the principles of Newton's laws of motion, as it explains how forces cause changes in an object's state of motion.

Unbalanced forces can also be observed in everyday activities such as pushing a car, throwing a ball, or pulling an object. Understanding these forces is crucial for comprehending the behaviour of objects in motion and predicting their motion. By analyzing the size and direction of these forces, we can determine the resulting acceleration and change in velocity of the objects.

The motion of objects without net force

Newton's laws of motion explain the relationship between a physical object and the forces acting upon it. Newton's first law of motion, also known as the law of inertia, states that an object will remain at rest or continue moving at a constant velocity unless acted upon by an unbalanced force. This means that if there is no net force acting on an object, its state of motion will not change. For example, a book on a table will stay there unless it is moved because the forces acting on it, such as gravity pulling it down and the table pushing it up, are balanced.

Newton's second law of motion defines a force to be equal to the change in momentum (mass times velocity) per change in time. This can be represented by the equation F=ma, where F is the force, m is the mass, and a is the acceleration. The second law talks about changes in momentum and can be used to determine the new values of velocity and mass if the force is known.

The motion of objects without a net force can be observed in real-world situations. For example, a spacecraft moving in the vacuum of space, far from any planets or stars, will continue moving at the same speed and in the same direction as there are no forces acting on it to slow it down or change its course. Similarly, a car on cruise control on a straight road can be considered to have zero net force when the force of wind resistance is matched by the power of the engine, resulting in a constant velocity.

While Newton's laws provide a foundation for understanding the motion of objects, the conceptual underpinnings of quantum physics differ significantly from classical physics. Quantum mechanics focuses on calculating the probability of specific measurement results rather than treating position, momentum, and energy as inherent properties of an object. The Ehrenfest theorem provides a connection between quantum expectation values and Newton's second law, but it is inexact due to the fundamental differences between quantum and classical physics.

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