The Newton's Laws of Motion are three fundamental principles that describe the motion of objects. They were formulated by the famous physicist Sir Isaac Newton and form the foundation for the understanding of classical mechanics.

For understanding Newton's Laws of Motion we need three following physical quantities:

The acceleration \( \class{red}{a} \) specifies how fast the velocity of a body changes. It is measured in the unit \( \mathrm{m}/\mathrm{s}^2 \) (meters per square second). If a body experiences an acceleration of \( 5 \, \mathrm{m}/\mathrm{s}^2 \), then its velocity increases by \( 5 \, \mathrm{m}/\mathrm{s} \) every second.

The mass \( \class{brown}{m} \) is a property of the body and is measured in \( \mathrm{kg} \) (kilograms). This physical quantity indicates how much matter a body consists of.

The force \( F \) is a physical quantity relating the mass \( \class{brown}{m} \) of the body to its acceleration \( a \). Force is measured in units of \( \mathrm{N} \) (Newton). If you apply \( F = 1 \, \mathrm{N} \) force to a one kilogram body, it will accelerate with \( 1 \, \mathrm{m}/\mathrm{s}^2 \). The velocity of the body increases with time as long as this force acts on it!

Newton's First Law of Motion

Newton's First Law of Motion states that a body remains at rest or moves with constant velocity \( \class{blue}{v} \) as long as no external forces act on the body. In other words, a body maintains its state of motion as long as no forces act on it.

If no resultant force \( F \) is applied to a body at rest, it remains at rest: \( \class{blue}{v} = 0 \).

If no resultant force acts on a moving body: \( F = 0 \), then this body will continue to move at constant velocity: \( \class{blue}{v} = \mathrm{const} \). For example, if a vehicle is traveling at \( \class{blue}{v} = 5 \, \mathrm{m}/\mathrm{s} \) and this velocity remains the same, then by Newton's 1st axiom there is no resultant force acting on the vehicle.

The second law states that the acceleration \( \class{red}{a} \) of a body is proportional to the resulting force \( F \) and inversely proportional to the mass \( \class{brown}{m} \) of the body. We can express this principle with an easy to remember formula:

Formula anchor$$ \begin{align} F = \class{brown}{m} \, \class{red}{a} \end{align} $$

The greater the force \( F \) acting on the body, the greater the acceleration \( \class{red}{a} \). If you double the force on the body, then the body will accelerate twice as fast.

A body with a larger mass \( \class{brown}{m} \) is more difficult to accelerate than a lighter body. Thus, if an equal force is applied to a heavy body and a light body, the light body will be accelerated more.

The 2nd Newton axiom can also be expressed in terms of the momentum \( p = \class{brown}{m} \, \class{blue}{v} \) of a body. The momentum is the product of the mass of the body and its velocity. The velocity of an accelerated body changes with time. For a uniform acceleration we can write: \( \class{red}{a} = \Delta v / \Delta t \). In other words: Within the time span \( \Delta t \) the velocity has changed by \( \Delta v \).

The change in velocity naturally leads to a change in momentum:

Formula: Change of momentum with time

Formula anchor$$ \begin{align} \Delta p ~=~ \class{brown}{m} \, \Delta v \end{align} $$

We assume that the mass remains the same during the velocity change. If we now replace \( \Delta v \) with \( \Delta v = \class{red}{a} \, \Delta t \), we get the following equation:

Formula: Change of momentum expressed with acceleration

Formula anchor$$ \begin{align} \Delta p ~=~ \class{brown}{m} \, \class{red}{a} \, \Delta t \end{align} $$

According to the 2nd Newton law, \( \class{brown}{m} \, \class{red}{a} \) corresponds to a resultant force \( F \):

Change of momentum expressed with force and time

Formula anchor$$ \begin{align} \Delta p ~=~ F \, \Delta t \end{align} $$

If we only bring \( \Delta t \) to the left side of the equation, we get the 2nd Newton's law, which is expressed with the temporal change of momentum:

2nd Newton law expressed with momentum

Formula anchor$$ \begin{align} \frac{\Delta p}{\Delta t} ~=~ F \end{align} $$

This formula states that a force acting on a body causes the momentum of the body to change.

Newton's Third Law of Motion

Newton's third law states that every force \( F_{\text{actio}} \) (action) is followed by an equal and opposite force \( F_{\text{reactio}} \) (reaction). If a body A exerts a force \( F_{\text{actio}} \) on another body B (action), the body B will exert an equal but opposite force \( F_{\text{reactio}} \) on the body A (reaction). In other words, forces always occur in pairs and act on different bodies.

Newton's 3rd law of axiom can be expressed by the following formula:

Every force (Actio) generates a counterforce (Reactio)

Formula anchor$$ \begin{align} F_{\text{actio}} ~=~ -F_{\text{reactio}} \end{align} $$

Statement of Newton's 3rd law of axiom

Each force \( F_{\text{actio}} \) causes an equal, oppositely directed counterforce \( F_{\text{reactio}} \).

Example of force and counterforce

You apply a force of \( F_{\text{actio}} = 10 \, \mathrm{N} \) to a box so that it is moved to the right. According to Newton's 3rd law, there is an equal counterforce \( F_{\text{reactio}} = 10 \, \mathrm{N} \), which has the same magnitude and is exerted by the box on you and in the opposite direction (to the left).

+ Perfect for high school and undergraduate physics students + Contains over 500 illustrated formulas on just 140 pages + Contains tables with examples and measured constants + Easy for everyone because without vectors and integrals