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Dynamics: Newton laws and the reign of parabolas

The laws

The goal of Newton, in its book “ Philosophiæ naturalis principia mathematica”, written in 1687, was to find some simple laws allowing to explain the motion of bodies. In physics, a law is the building block of a theory. It is a statement, often mathematical, which can not be proven from other parts of the theory but can be used to make predictions. A theory is treated as correct as long as its building laws do not lead to predictions which contradicts experiment. It is quite astonishing that, accepting the three relatively simple laws of Newton, we will be able to explain and predict almost all of the simple case of motions we see around us. For exemple, we will be able to understood very well the orbit of the planets around the Sun.

The whole understanding of Newton’s theory relies under the notion of forces. In his laws, Newton defines the forces and states that forces are what drives the motions of all the bodies in the Universe. The three laws can then be stated as follows:

First law (principle of inertia): A body under the influence of no forces moves in straight uniform motion that is whith a constant velocity vector $\vec{v}$. We say that the body is free.
Second law (fundamental principle of dynamics): If a body stops being free, its accelerates $\vec{a}=\dot{\vec{v}}\neq0$. This acceleration is due to the presence of forces, which are vectors $\vec{F}_i$. The sum of all the forces act to accelerate the body such that $\vec{a}=\sum_i\vec{F}_i/m$.
Third law (action/reaction principle): If a body $A$ exert a force $\vec{F}_{A\to B}$ on a body $B$, then the body $A$ must also witness a force coming from the body $B$, with identical amplitude but opposite direction. That is $\vec{F}_{A\to B}= -\vec{F}_{B\to A}$.

While we saw what was kinematics before, as the tools used to describe the motions, Newton laws talk about the dynamics i.e. they allow to predict how the motions will be or were in the past.

Let us now explore the laws and their consequences in greater detail.

First law

Newton vs Aristotle.

Second law

Newton’s second law is often written in the form

\[\sum_i \vec{F}_i = m\vec{a}\] \[\vec{a}=\frac{\sum_i \vec{F}_i}{m}\]

The linear momentum

\[\vec{p}=m\vec{v}\] \[\frac{\text{d}\vec{p}}{\text{d}t}=\sum_i \vec{F}_i\] \[\frac{\text{d}\vec{p}}{\text{d}t}=\vec{0}\]

Third law

\[\vec{F}_{A\to B}= -\vec{F}_{B\to A}\]

Gravity

\[\vec{P}=m\vec{g}\] \[\vec{F}_G=-\frac{GmM}{r^2}\vec{u}_r\] \[\left|\vec{g}\right|= \frac{GM}{R_T^2}\simeq 9.81 {\rm m.s}^{-2}\]

$\sum_i{\vec{F}_i}=0$, $\vec{R}=-m\vec{g}$.

$\vec{g}$ on other planets.

Application: the reign of parabolas

\[\sum_i\vec{F}=\vec{P}\] \[m\vec{a}= m\vec{g}\] \[\ddot{z}= -g\]

Application: A falling ant

\[\sum_i \vec{F}= \vec{R} + \vec{P}\]

$\vec{R} = kv^2 \vec{u}_z$ and $\vec{P}=-mg \vec{u}_z$, such that $\sum_i \vec{F}=(kv^2-mg)\vec{u}_z$.

Additionaly, Newton second law reads

\[\sum_i \vec{F}= m a \vec{u}_z\]

Such that, projecting on the $z$ axis, we obtain

\[ma = kv-mg\] \[\ddot{z} = \frac{k}{m}\dot{z}^2 - g\]

The terminal velocity $\ddot{z}=0$:

\[v_{\rm t} = \dot{z}_t = \sqrt{\frac{gm}{k}}\]

Solving the differential equation

\[z(t)=\]