Newton's law of Motion

Newton's Law of Motion :
Newton's laws of motion are three physical laws that, together, laid the foundation for
classical mechanics . They describe the relationship between a body and the forces acting upon it, and its motion in response to those forces. They have been expressed in several different ways, over nearly three centuries, and can be summarised as follows.
First law:
In an inertial reference frame , an object either remains at rest or continues to move at a constant velocity , unless acted upon by a net force.
Second law:
In an inertial reference frame, the sum of the forces F on an object is equal to the mass m of that object multiplied by the acceleration a of the object: F =
m a .
Third law:
When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body.
The three laws of motion were first compiled by
Isaac Newton in his Philosophic Naturalist Principia (Mathematical Principles of Natural Philosophy ), first published in 1687.
Newton used them to explain and investigate the motion of many physical objects and systems. For example, in the third volume of the text, Newton showed that these laws of motion, combined with his law of universal gravitation , explained Kepler's laws of planetary motion .

Newton's laws of motion are three physical laws that, together, laid the foundation for
classical mechanics . They describe the relationship between a body and the forces acting upon it, and its motion in response to those forces. They have been expressed in several different ways, over nearly three centuries, and can be summarised as follows.
First law:
In an inertial reference frame , an object either remains at rest or continues to move at a constant velocity , unless acted upon by a net force.
Second law:
In an inertial reference frame, the sum of the forces F on an object is equal to the mass m of that object multiplied by the acceleration a of the object: F =
m a .
Third law:
When one body exerts a force on a second body, the second body simultaneously exerts a force equal in magnitude and opposite in direction on the first body.
The three laws of motion were first compiled by
Isaac Newton in his Philosophic Naturalist Principia Mathematica (Mathematical Principles of Natural Philosophy ), first published in 1687.
Newton used them to explain and investigate the motion of many physical objects and systems.  For example, in the third volume of the text, Newton showed that these laws of motion, combined with his law of universal gravitation , explained Kepler's laws of planetary motion .

Newton's laws hold only with respect to a certain set of frames of reference called
Newtonian or inertial reference frames . Some authors interpret the first law as defining what an inertial reference frame is; from this point of view, the second law only holds when the observation is made from an inertial reference frame, and therefore the first law cannot be proved as a special case of the second. Other authors do treat the first law as a corollary of the second. The explicit concept of an inertial frame of reference was not developed until long after Newton's death.
In the given interpretation mass , acceleration ,
momentum , and (most importantly) force are assumed to be externally defined quantities. This is the most common, but not the only interpretation of the way one can consider the laws to be a definition of these quantities.
Newtonian mechanics has been superseded by
special relativity , but it is still useful as an approximation when the speeds involved are much slower than the speed of light.
***The first law states that if the net force (the vector sum of all forces acting on an object) is zero, then the velocity of the object is constant. Velocity is a vector quantity which expresses both the object's speed and the direction of its motion; therefore, the statement that the object's velocity is constant is a statement that both its speed and the direction of its motion are constant.
The first law can be stated mathematically when the mass is a non-zero constant, as,
Consequently,
An object that is at rest will stay at rest unless a force acts upon it.
An object that is in motion will not change its velocity unless a force acts upon it.
This is known as uniform motion . An object
continues to do whatever it happens to be doing unless a force is exerted upon it. If it is at rest, it continues in a state of rest (demonstrated when a tablecloth is skilfully whipped from under dishes on a tabletop and the dishes remain in their initial state of rest). If an object is moving, it continues to move without turning or changing its speed. This is evident in space probes that continually move in outer space. Changes in motion must be imposed against the tendency of an object to retain its state of motion. In the absence of net forces, a moving object tends to move along a straight line path indefinitely.
Newton placed the first law of motion to establish frames of reference for which the other laws are applicable. The first law of motion postulates the existence of at least one
frame of reference called a Newtonian or inertial reference frame , relative to which the motion of a particle not subject to forces is a straight line at a constant speed.  Newton's first law is often referred to as the law of inertia . Thus, a condition necessary for the uniform motion of a particle relative to an inertial reference frame is that the total net force acting on it is zero.
In this sense, the first law can be restated as:
“In every material universe, the motion of a particle in a preferential reference frame Φ is determined by the action of forces whose total vanished for all times when and only when the velocity of the particle is constant in Φ. That is, a particle initially at rest or in uniform motion in the preferential frame Φ continues in that state unless compelled by forces to change it."

****The second law states that the rate of change of momentum of a body, is directly proportional to the force applied and this change in momentum takes place in the direction of the applied force.
The second law can also be stated in terms of an object's acceleration. Since Newton's second law is only valid for constant-mass systems,  it can be taken outside the
differentiation operator by the constant factor rule in differentiation . Thus,
where F is the net force applied, m is the mass of the body, and a is the body's acceleration. Thus, the net force applied to a body produces a proportional acceleration. In other words, if a body is accelerating, then there is a force on it.
Consistent with the first law, the time derivative of the momentum is non-zero when the momentum changes direction, even if there is no change in its magnitude; such is the case with uniform circular motion. The relationship also implies the conservation of momentum : when the net force on the body is zero, the momentum of the body is constant. Any net force is equal to the rate of change of the momentum.
Any mass that is gained or lost by the system will cause a change in momentum that is not the result of an external force. A different equation is necessary for variable-mass systems (see below ).
Newton's second law requires modification if the effects of special relativity are to be taken into account, because at high speeds the approximation that momentum is the product of rest mass and velocity is not accurate.
Impulse
An impulse J occurs when a force F acts over an interval of time Δt , and it is given by since force is the time derivative of momentum, it follows that
This relation between impulse and momentum is closer to Newton's wording of the second law.
Impulse is a concept frequently used in the analysis of collisions and impacts.
Variable-mass systems
Main article: Variable-mass system
Variable-mass systems, like a rocket burning fuel and ejecting spent gases, are not closed and cannot be directly treated by making mass a function of time in the second law;  that is, the following formula is wrong:
The falsehood of this formula can be seen by noting that it does not respect Galilean invariance : a variable-mass object with F = 0 in one frame will be seen to have F ≠ 0 in another frame. The correct equation of motion for a body whose mass m varies with time by either ejecting or accreting mass is obtained by applying the second law to the entire, constant-mass system consisting of the body and its ejected/accreted mass; From this equation one can derive the equation of motion for a varying mass system, for example, the
Tsiolkovsky rocket equation . Under some conventions, the quantity u  d m /d t on the left-hand side, which represents the advection of
momentum , is defined as a force (the force exerted on the body by the changing mass, such as rocket exhaust) and is included in the quantity F . Then, by substituting the definition of acceleration, the equation becomes F = m a .

****Newton's third law
An illustration of Newton's third law in which two skaters push against each other. The first skater on the left exerts a normal force N12 on the second skater directed towards the right, and the second skater exerts a normal force N21 on the first skater directed towards the left.
The magnitudes of both forces are equal, but they have opposite directions, as dictated by Newton's third law.
The third law states that all forces between two objects exist in equal magnitude and opposite direction: if one object A exerts a force F A on a second object B , then B simultaneously exerts a force F B on A , and the two forces are equal in magnitude and opposite in direction: F A = − FB .
The third law means that all forces are
interactions between different bodies, or different regions within one body, and thus that there is no such thing as a force that is not accompanied by an equal and opposite force. In some situations, the magnitude and direction of the forces are determined entirely by one of the two bodies, say Body A; the force exerted by Body A on Body B is called the "action", and the force exerted by Body B on Body A is called the "reaction". This law is sometimes referred to as the action-reaction law , with FA called the "action" and F B the "reaction". In other situations the magnitude and directions of the forces are determined jointly by both bodies and it isn't necessary to identify one force as the "action" and the other as the "reaction". The action and the reaction are simultaneous, and it does not matter which is called the action and which is called reaction; both forces are part of a single interaction, and neither force exists without the other.
The two forces in Newton's third law are of the same type (e.g., if the road exerts a forward frictional force on an accelerating car's tires, then it is also a frictional force that Newton's third law predicts for the tires pushing backward on the road).
From a conceptual standpoint, Newton's third law is seen when a person walks: they push against the floor, and the floor pushes against the person. Similarly, the tires of a car push against the road while the road pushes back on the tires—the tires and road simultaneously push against each other. In swimming, a person interacts with the water, pushing the water backward, while the water simultaneously pushes the person forward—both the person and the water push against each other. The reaction forces account for the motion in these examples. These forces depend on friction; a person or car on ice, for example, may be unable to exert the action force to produce the needed reaction force.