AMOUNT OF MOTION: CONSERVATION LAW, CLASSICAL MECHANICS - PHYSICAL - 2024

The amount of motion or linear momentum, also known as momentum, is defined as a physical magnitude in the vector-type classification, which describes the movement that a body performs in mechanical theory. There are several types of mechanics that are defined in the amount of movement or momentum.

Classical mechanics is one of those types of mechanics and can be defined as the product of the mass of the body and the speed of movement at a given instant. Relativistic mechanics and quantum mechanics are also part of linear momentum.

There are various formulations for the amount of movement. For example, Newtonian mechanics defines it as the product of mass and velocity, while Lagrangian mechanics requires the use of self-adjoint operators defined on a vector space in an infinite dimension.

The momentum is governed by a conservation law, which states that the total momentum of any closed system cannot be altered and will always remain constant over time.

Law of conservation of momentum

In general terms, the law of conservation of momentum or momentum states that, when a body is at rest, it is easier to associate inertia with mass.

Thanks to the mass, we obtain the magnitude that will allow us to remove a body at rest and, in the case that the body is already in motion, the mass will be a determining factor when changing the direction of the speed.

This means that, depending on the amount of linear movement, the inertia of a body will depend on both the mass and the speed.

The momentum equation expresses that the momentum corresponds to the product of the mass and the velocity of the body.

p = mv

In this expression p is the momentum, m is the mass and v is the velocity.

Classical mechanics

Classical mechanics studies the laws of the behavior of macroscopic bodies at speeds much lower than that of light. This momentum mechanic is divided into three types:

Newtonian mechanics

Newtonian mechanics, named after Isaac Newton, is a formula that studies the motion of particles and solids in three-dimensional space. This theory is subdivided into static mechanics, kinematic mechanics and dynamic mechanics.

The statics deals with the forces used in a mechanical equilibrium, the kinematics studies the movement without taking into account the result of the same and the mechanics studies both the movements and the results of the same.

Newtonian mechanics is used primarily to describe phenomena that occur at a speed much slower than the speed of light and on a macroscopic scale.

Langragian and Hamiltonian Mechanics

Langrian mechanics and Hamiltonian mechanics are very similar. Langragian mechanics is very general; for that reason, its equations are invariant with respect to some change in the coordinates.

This mechanics provides a system of a certain amount of differential equations known as equations of motion, with which it can be inferred how the system will evolve.

On the other hand, Hamiltonian mechanics represents the momentary evolution of any system through first-order differential equations. This process allows the equations to be much easier to integrate.

Continuous media mechanics

Continuous media mechanics is used to provide a mathematical model where the behavior of any material can be described.

Continuous media are used when we want to find out the momentum of a fluid; in this case the momentum of each particle is added.

Relativistic mechanics

The relativistic mechanics of quantity of motion - also following Newton's laws - states that, since time and space exist outside of any physical object, Galilean invariance takes place.

For his part, Einstein maintains that the postulation of the equations does not depend on a frame of reference but accepts that the speed of light is invariable.

In the momentum, relativistic mechanics works similar to classical mechanics. This means that this magnitude is greater when it refers to large masses, which move at very high speeds.

In turn, it indicates that a large object cannot reach the speed of light, because eventually its momentum would be infinite, which would be an unreasonable value.

Quantum mechanics

Quantum mechanics is defined as an articulation operator in a wave function and which follows Heinsenberg's uncertainty principle.

This principle sets limits on the precision of the moment and the position of the observable system, and both can be discovered at the same time.

Quantum mechanics uses relativistic elements when addressing various problems; this process is known as relativistic quantum mechanics.

Relationship between momentum and momentum

As mentioned previously, the momentum is the product of the velocity and the mass of the object. In the same field, there is a phenomenon known as momentum, which is often confused with momentum.

The momentum is the product of the force and the time during which the force is applied and is characterized by being considered a vector quantity.

The main relationship between momentum and momentum is that the momentum applied to a body is equal to the change in momentum.

In turn, since the momentum is the product of force and time, a certain force applied in a given time causes a change in the momentum (without taking into account the mass of the object).

Momentum exercise

A baseball of mass 0.15 kg is moving with a speed of 40 m / s when it is hit by a bat that reverses its direction, acquiring a speed of 60 m / s, what average force did the bat exert on the ball if it was in contact with this 5 ms ?.

Solution

Data

m = 0.15 kg

vi = 40 m / s

vf = - 60 m / s (the sign is negative since it changes the direction)

t = 5 ms = 0.005 s

Δp = I

pf - pi = I

m.vf - m.vi = Ft

F = m. (Vf - vi) / t

F = 0.15 kg. (- 60 m / s - 40 m / s) / 0.005 s

F = 0.15 kg. (- 100 m / s) / 0.005 s

F = - 3000 N

References

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