Newton’s Laws of Motion

Newton’s laws of motion are a set of three physical laws that together form the foundation of classical mechanics. They basically connect the motion of matter with force applied.

Let us try to understand these laws in detail..

Before Newton

To understand how anything came to be it’s important to understand what was before. Hence, to understand Newton’s Laws of Motion it is important to understand what laws were in place before. Why were Newton’s Laws needed? What were the problems with the laws before? Answering these questions will lead to how Newton’s Laws of Motion came to be, and what we stand to benefit by learning about them.

Aristotle’s Fallacy

Fallacy: a mistaken belief, especially one based on unsound arguments.

Most of early physics was based on assumptions and didn’t usually have backing arguments, proofs or data, to strengthen its claims. Most of these claims would be based on observation of the environment, and the physicists surroundings, but there is more to nature than meets the eye. Aristotle was of the belief that a continual impetus was needed to maintain motion. In simple terms, you will need to keep on applying force, to ensure that motion is continued. 

For example, he explains the motion of an arrow by stating that the air displaced by a moving arrow moves back to the base of the arrow, pushing it, providing it force, to continue the motion of the arrow. But obviously this is disproved, by shooting an arrow in vacuum (absence of air), and observing that motion is maintained as well.

So is continuous force really required to maintain motion? But, then why do things eventually stop moving if an initial force is given and then removed?

Galileo’s Law of Inertia

Inertia: property of a body to resist change

Galileo often stated to be the ‘father of modern science’. While Galileo’s predecessors were wrong on matters of motion, his ideas of it are still correct in part. Galileo’s Law of Inertia arises from two simple observations:

1. The speed of ball decreases when it rises up a slope
2. The speed of a ball increases when it falls down a slope.

Using these facts Galileo designed an exp)eriment shown below.

 Two inclined planes were joined to form a bowl-like structure. It is observed that when a ball is left from a point at the top of one plane, the ball moves on the path shown and reaches a height equal to the height we released the ball from. When the planes are separated a similar result is observed. What would happen if one of the inclined planes is on the surface? Theoretically speaking the ball will keep on moving without any change in its speed, that is the velocity of the ball will be constant on a plane surface.

Galileo through other observations, assumptions and shrewd thinking stated his law of inertia as: ‘a body in a state of rest will remain in a state of rest and a body in motion with constant velocity will remain in motion with same constant velocity’

But this still leaves the question of why a moving body eventually comes to rest in a practical situation? Perhaps Newton’s Laws could help answer the question.

Newton’s First Law of Motion

Following this, Newton built on Galileo’s idea of inertia and hence stated his First Law of Motion as: ‘a body will maintain its state of rest or motion with constant velocity unless acted upon by a net external force’ 

With this Newton introduces to us the term force. In simple terms it means a push or a pull that strives to bring about a movement in the direction it was applied in. The term ‘net’ used in this context means that forces add up and can cancel each other. Think of it like if too equally strong individuals push a box from either side, the box doesn’t move, or if there is a tug of war between two equally powerful teams, the central point doesn’t move.

And with Newton’s First Law we can think about answering an unanswered question. Why does a moving body eventually stop? Newton’s Law states that there must be an external force to account for the change in motion. And this is in fact the case, as friction between the body and surface it moves on results in the decrease of velocity and the eventual stopping of motion of the body. 

Newton’s Second Law of Motion

But how do we measure force, and what is the mathematical definition of force? Newton’s Second Law of Motion comes to our rescue. It states: ‘the rate of change of momentum is directly proportional to the applied force, and is in the direction the force was applied in’

Momentum: a measured quantity specific for a body in motion which is the mass of the body multiplied with its velocity.

The mathematics comes to look like:
F∝ Δp/Δt       
where,
F is the applied force
Δp is the change in momentum
And, Δt is the change in time.

Therefore, Δp/Δt comes out to be the rate of change in momentum

Since, p = mv ⇒ Δp = Δ(mv) = mΔv
[Assuming, that mass remains constant.]
where v is the velocity of the body
m is the mass of the body

We have:
F ∝ Δp/Δt = Δ(mv)/Δt = m(Δv/Δt) = ma
F ∝ ma
F = kma
where,
v/t = a is the acceleration of the moving body
k is the proportionality constant

[A proportionality constant is introduced to remove proportionality. This converts it into an equation. Proportionality constants can take any numeric value to ensure the equation is correct]

For smaller changes in momentum and velocity where,
dp is a small change in momentum
dv is a small change in velocity
dt is a small change in time

Since, p = mv ⇒ dp = d(mv) = mdv
[Assuming, that mass remains constant.]

We have:
F ∝ dp/dt = (mdv)/dt = m(dv/dt) = ma
F ∝ ma
F = kma
where, dv/dt = a is the acceleration of the moving body

To measure force we introduce the unit ‘Newton’, where
1 Newton = 1kg x 1m/s² 

If we compare this definition of a Newton with the equation above, that is F=kma, we see that the proportionality constant k takes the value of 1.

Giving the equation for force as:
F = ma

If we can’t assume that mass is constant, a circumstance possible at relativistic speeds (v>>c, where c is the speed of light), the most accurate mathematical form of Newton’s Second Law comes to be:
F = dp/dt

Newton’s Third Law of Motion

Newton’s Third Law becomes important to explain some physical phenomena. Simple questions like, ‘How do we walk?’, ‘How does a rocket move forward?’, etc. We walk by applying force upon the ground, especially through the balls of our feet. A rocket moves by expanding hot gas from its boosters. But these forces are applied in directions that are opposite to the resultant movement. What forces cause the movement then? 

Newton’s Third Law, perhaps one of the most quoted laws helps us with this, with the statement being: ‘every action has an equal and directionally opposite reaction’

Action and Reaction are nothing but special names for normal forces. Using this law, the above examples can be explained easily. The ground applies an equal force on us in the opposite direction. The expanding gas pushes the rocket upwards. In both cases the reaction is responsible for moving the body. 

Newton’s Laws Now

Newton’s Three Laws were originally described in his 1687 work, Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), and to this date can be used to explain and predict a wide range of phenomena at everyday life scales and speeds. Combined with Newton’s Law of Gravitation, excellent approximations can be made up of planetary motions.

However these laws break down for bodies which are very small, moving too fast, or are present inside a high gravitational field. To explain such phenomena more sophisticated theories such as general theory of relativity, quantum field theory, etc. are needed.