# Newton's laws of motion

In this page, I am going to discuss Newton's laws of motion as they apply to **point masses** (or **point particles**).

## Contents

**Newton's First Law (Law of Inertia)**[edit]

*In an inertial reference frame, a point mass remains at rest or continues to move uniformly unless acted upon a force* **or equivalently** *any particle persists on its own state of rest or uniform motion unless it is compelled to change that state by an external force.*

Where uniform motion means that the derivative of the position vector function is a constant vector, with respect our reference frame. Mathematically:

where is a non time-varying vector.

The first Law of Newton is unintuitive as it does not correspond to our every day experiences. For Instance, we never see an object moving with constant veclocity as this state requires the asbense of forces. This condition is mathematically expressed as:

where is i-th force, and N is the total number of forces acting on the point particle of our interest.

Examples of forces that prevent objects from undergoing uniform motion in our world are:

- Graviatational forces between objects (obeying the law of Universal Gravitation)

- Electrostatic forces between charged particles (obeying Coulomb's Law)

- the Lorentz Force (exerted on a charged particle when moving through a magnetic field)

- Frictional Forces

- Forces by gases when moving through them (example: The atmosphere)

**Newton's Second Law**[edit]

*For any particle of non zero mass m, the vector summation of all the forces acting on the particle is always equal to the mass m times the body's acceleration.*

This law is mathematically expressed as:

Many choose to write the previous equation as follows:

**Relation with momentum**

The First equation can be written as:

**Newton's second law as a differential equation**

Forces in classical mechanics are depended from the position of a particle in space (Example: Gravitational Forces), from the velocity of a particle moving in space (Example: Forces exerted by gases or the Lorentz force) and there can be time-varying forces

(, where **g(t)** is some function of time with dimensions ).

When the second law of Newton is written in the form

To illustrate that, consider the following cases:

**Horizontal Motion with Linear Drag:**

Consider a particle S moving horizontally in a linearly resistive medium (where at ) and also assume that the force exerted by the medium on the particle is . The second law of Newton will completely define the trajectory of the particle. Therefore:

Notice that the, so after a sufficient amount of time (theoretically infinite) the particle will come to a stop. Based on the particles velocity we can find the position since it is true that: Therefore, the position function will be:

Notice that

**Force proportional to the distance (Linear motion)**

Suppose we have a particle S and a Force is exerted on it where

Once again, The second law will determine the particle's position function:

_{1}and C

_{2}are arbitary constants that are determined by the inital conditions. Therefore:

**Graph of the particle's velocity****Graph of position:**

**Constant Force**

This time, the force exerted on an arbitary particle S is ** (Assume ). The second law of Newton gives us immediately the second derivative of position.**

**m**is the mass of the particle. Based on the definition of velocity we get: