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Newton's Laws of Motion

Important Formula

Formula: 2. Newton's laws of motion

$$F = \class{brown}{m} \, \class{red}{a}$$ $$F = \class{brown}{m} \, \class{red}{a}$$ $$\class{brown}{m} ~=~ \frac{F}{\class{red}{a}}$$ $$\class{red}{a} = \frac{F}{\class{brown}{m}}$$

Rearrange formula

What do the formula symbols mean?

Force

$$ \boldsymbol{F} $$ Unit $$ \mathrm{N} = \frac{\mathrm{kg} \, \mathrm{m}}{\mathrm{s}^2} $$

Force that accelerates the body of mass $m$.

Mass

$$ \class{brown}{m} $$ Unit $$ \mathrm{kg} $$

The mass is the proportionality constant in Newton's second law of motion. It defines how effective a body can be accelerated.

Acceleration

$$ \class{red}{\boldsymbol a} $$ Unit $$ \frac{\mathrm m}{\mathrm{s}^2} $$

The acceleration tells by how fast the velocity of the body changes per second.

Table of contents

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.

Both at the time $t_1$ and at a later time $t_2$ the body has the same velocity $ \class{blue}{v} $ when there are no forces acting on the body.

Statement of Newtons First Law of Motion

A body remains at rest or moves at a constant velocity as long as no forces act on it.

Newton's Second Law of Motion

While the 1st Newton's law describes unaccelerated motion, the Newton's 2nd Law deals with accelerated motion.

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:

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

Newtons Second Law of Motion: Force Accelerates a Body

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.

Statement of Newtons Second Law of Motion

The acceleration of a body is proportional to the resulting force.

Example: Calculate acceleration of a body

A force of 10 newtons acts on a body with a mass of 5 kg. How large is the resulting acceleration that the body experiences?

To do this, rearrange Newton's second law 1 with respect to the acceleration and insert the given values:

$$ \begin{align} \class{red}{a} ~&=~ \frac{F}{\class{brown}{m}} \\\\
~&=~ \frac{10 \, \mathrm{N}}{ \class{brown}{ 5 \, \mathrm{kg} } } \\\\
~&=~ 2 \, \frac{ \mathrm m }{ \mathrm{s}^2} \end{align} $$

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:

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

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:

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

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

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

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:

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

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

Example: Change of momentum after an application of force

A force of 10 newtons acts on a body within 3 seconds. What change in momentum has the body experienced?

To do this, rearrange formula 6 with respect to the momentum change $ \Delta p $ and insert the given values:

$$ \begin{align} \Delta p ~&=~ F \, \Delta t \\\\
~&=~ 10 \, \mathrm{N} \cdot 3 \, \mathrm{s} \\\\
~&=~ 30 \, \frac{ \mathrm{kg} \cdot \mathrm{m} }{ \mathrm{s}} \end{align} $$

If we can assume that the mass of the body has not changed during this time interval, this change in momentum has caused the velocity of the body to change by $ 30 \, \frac{ \mathrm{m} }{ \mathrm{s}} $.

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:

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

Example: Rocket takes off

A rocket exerts a force $F_{\text{actio}}$ on the fuel. According to Newton's third law, the ejected fuel exerts a counterforce $F_{\text{reactio}}$ on the rocket and it lifts off. It is very important to understand that the force and the counterforce do NOT act on the same body! If they did, they would cancel each other out and the resulting force would be zero - the rocket would not take off at all.

Newtons Third Law of Motion: Force on the Fuel and Counterforce on the Rocket

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).

When is a body in equilibrium of forces?

A point of mass is in force equilibrium when the sum of all forces $F_i$ acting on the body adds up to zero:

$$ \begin{align} F ~=~ F_1 + F_2 + ~... ~=~ 0 \end{align} $$

When do Newton's laws fail?

Newtonian mechanics does not work when you want to describe accelerated or curved motion, even if you are in an accelerated reference frame yourself. In such cases, so-called inertial forces occur, which have no real cause and therefore no opposing force ("reactio"). In summary, Newton's axioms do not apply in non-inertial systems.

They also fail at too high speeds (see theory of relativity). But also in the microcosm when describing quantum mechanical objects like electrons.