Comprehensive Engineering Physics: Units, Vectors, Kinematics, and Dynamics

Foundations of Physics, Standards, and Units

Physics (physique) is the fundamental science (science fondamentale) concerned with discovering the basic principles of the universe (univers). It serves as the foundation upon which all other physical sciences (sciences physiques) are constructed, characterized by the inherent simplicity of its fundamental concepts (concepts fondamentaux). The primary objective of physics is to model nature by identifying the limited number of fundamental laws (lois fondamentales) that govern natural phenomena (phénomènes naturels). These laws are utilized to develop theories (théories) capable of predicting the outcomes of future experiments (expériences). Mathematics (mathématiques) acts as the essential bridge between theoretical frameworks and experimental observations, providing the language through which physical laws are expressed.

To quantitatively describe natural phenomena, accurate measurements (mesures) are required, and each measurement corresponds to a physical quantity (grandeur physique). Reliable measurements require established, standardized units (unités standardisées). An acceptable standard of measurement (étalon de mesure) must be readily accessible, possess properties that can be measured with high reliability, yield identical results when applied by anyone anywhere in the world, and remain invariable over time. Standardized systems are officially established and maintained by international authoritative bodies. The principal system utilized globally is the Système International (SI) (Système international d'unités), which was formally adopted in 1960 by an international committee.

A physical quantity (grandeur physique) is characterized by an operational definition (définition opérationnelle), a numerical value (valeur numérique) or magnitude (magnitude / grandeur), and a corresponding physical unit (unité physique). Many physical quantities also possess a spatial direction (direction spatiale), dividing quantities into scalars (scalaires) and vectors (vecteurs). Magnitudes are conventionally expressed using a variable symbol accompanied by scientific notation (notation scientifique). For instance, the mass (masse) of a coin is represented as $m = 0.042\,\text{kg}$.

The SI system is anchored by seven fundamental physical quantities (grandeurs physiques fondamentales) and their base units (unités de base): length (longueur) is measured in meters ($\text{m}$) (mètres), mass (masse) in kilograms ($\text{kg}$) (kilogrammes), time (temps) in seconds ($\text{s}$) (secondes), temperature (température) in Kelvin ($\text{K}$) (kelvin), electric current (courant électrique) in Ampere ($\text{A}$) (ampère), luminous intensity (intensité lumineuse) in candela ($\text{cd}$) (candela), and amount of substance (quantité de matière) in moles ($\text{mol}$) (moles).

In classical mechanics (mécanique classique), all phenomena can be described using only three fundamental quantities: length, mass, and time. All other mechanical quantities, such as velocity (vitesse), force (force), and energy (énergie), are derived quantities (grandeurs dérivées) expressible as combinations of these three base quantities.

Length (longueur) represents the measure of distance (distance) between two points in space. The SI base unit of length is the meter ($\text{m}$) (mètre), defined precisely as the distance traveled by light in a vacuum (vide) during a time interval of $\frac{1}{299\,792\,458}$ of a second. Measured lengths across the physical universe encompass an enormous dynamic scale:

• Distance from the Earth to the most remote known quasar: $1.4 \times 10^{26}\,\text{m}$

• Distance from the Earth to the most remote normal galaxies: $9 \times 10^{25}\,\text{m}$

• Distance from the Earth to the nearest large galaxy (Andromeda): $2 \times 10^{22}\,\text{m}$

• Distance from the Sun to the nearest star (Proxima Centauri): $4 \times 10^{16}\,\text{m}$

• One light-year (année-lumière): $9.46 \times 10^{15}\,\text{m}$

• Mean orbit radius of the Earth about the Sun: $1.50 \times 10^{11}\,\text{m}$

• Mean distance from the Earth to the Moon: $3.84 \times 10^{8}\,\text{m}$

• Distance from the equator to the North Pole: $1.00 \times 10^{7}\,\text{m}$

• Mean radius of the Earth: $6.37 \times 10^{6}\,\text{m}$

• Typical altitude of a satellite orbiting the Earth: $2 \times 10^{5}\,\text{m}$

• Length of a football field: $9.1 \times 10^{1}\,\text{m}$

• Length of a housefly: $5 \times 10^{-3}\,\text{m}$

• Size of smallest dust particles: $\sim 10^{-4}\,\text{m}$

• Size of cells of most living organisms: $\sim 10^{-5}\,\text{m}$

• Diameter of a hydrogen atom: $\sim 10^{-10}\,\text{m}$

• Diameter of an atomic nucleus: $\sim 10^{-14}\,\text{m}$

• Diameter of a proton: $\sim 10^{-15}\,\text{m}$

Mass (masse) represents the measure of the quantity of matter (quantité de matière) within an object. The SI base unit of mass is the kilogram ($\text{kg}$) (kilogramme), historically established by the mass of a specific platinum-iridium cylinder maintained at the International Bureau of Weights and Measures.

Time (temps) is quantified in seconds ($\text{s}$) (secondes). The second is operationally defined in terms of the frequency of atomic transitions and oscillations of radiation emitted by a cesium atom ($\text{Cs-133}$).

Metric prefixes (préfixes métriques) represent decimal multipliers corresponding to specific powers of 10. These prefixes are combined with any base or derived unit name and symbol to scale quantities efficiently. For example, the prefix milli- corresponds to $10^{-3}$, such that $1\,\text{mm} = 10^{-3}\,\text{m}$ and $1\,\text{mg} = 10^{-3}\,\text{g}$.

The dimension (dimension) of a physical quantity denotes its fundamental physical nature. Dimensions are symbolized using capital letters in square brackets: length as $[L]$, mass as $[M]$, and time as $[T]$. Dimensional analysis (analyse dimensionnelle) serves as an indispensable tool for checking the mathematical and physical consistency of derived equations.

Coordinate Systems, Vectors, and Vector Algebra

Describing spatial positions and directional phenomena necessitates a coordinate system (système de coordonnées). A point in a two-dimensional plane can be represented either by Cartesian coordinates (coordonnées cartésiennes) $(x, y)$ or by polar coordinates (coordonnées polaires) $(r, \theta)$, where $r$ represents the radial distance (distance radiale) from the origin (origine) and $\theta$ denotes the angle (angle) measured with respect to the positive $x$-axis. The geometric transformations between Cartesian and polar systems follow standard right-angled trigonometry (trigonométrie du triangle rectangle):

$x = r \cos(\theta)$

$y = r \sin(\theta)$

$\tan(\theta) = \frac{y}{x}$

$r = \sqrt{x^2 + y^2}$

Physical quantities are divided fundamentally into scalars (scalaires) and vectors (vecteurs). A scalar quantity (grandeur scalaire) is completely specified by a single numerical value, an algebraic sign, and an appropriate unit. Scalars can be positive, negative, or zero. Examples include distance (distance), speed (vitesse scalaire), mass (masse), density (masse volumique / densité), energy (énergie), work (travail), power (puissance), temperature (température), and pressure (pression). A vector quantity (grandeur vectorielle) is completely characterized by a numerical magnitude (magnitude / norme), physical units, and a specific spatial direction (direction spatiale). Vector quantities include displacement (déplacement), velocity (vecteur vitesse), acceleration (accélération), force (force), momentum (quantité de mouvement), torque (moment de force / couple), and electric field (champ électrique). In print, vectors are indicated with boldface type or an overarching arrow, such as $\vec{A}$. The magnitude (norme / grandeur) of a vector is designated by an italic symbol $A$ or the modulus $\Vert\vec{A}\Vert$. The magnitude of a vector is strictly positive and carries physical units.

A vector in a plane can be broken down into rectangular components (composantes rectangulaires), which represent the projections (projections) of the vector along the coordinate axes. When the angle $\theta$ is measured counterclockwise from the positive $x$-axis, the components are calculated as:

$A_x = A \cos(\theta)$

$A_y = A \sin(\theta)$

Any vector can be translated parallel to itself without changing its physical properties, allowing the $y$-component vector to be translated to the tip of the $x$-component vector to construct a right-angled triangle (triangle rectangle) where the hypotenuse (hypoténuse) is the magnitude $A$. The algebraic sign of a component depends upon the quadrant in which the vector lies. The magnitude (norme) and orientation angle (angle d'orientation) are recovered from rectangular components via:

$A = \sqrt{A_x^2 + A_y^2}$

$\theta = \tan^{-1}\left(\frac{A_y}{A_x}\right)$

A unit vector (vecteur unitaire) is a dimensionless vector having a magnitude of exactly 1, used solely to point in a specified direction. The standard unit vectors pointing along the positive $x$, $y$, and $z$ axes are designated as $\hat{i}$, $\hat{j}$, and $\hat{k}$, where $\Vert\hat{i}\Vert = \Vert\hat{j}\Vert = \Vert\hat{k}\Vert = 1$. A two-dimensional vector is expressed in unit-vector notation as:

$\vec{A} = A_x \hat{i} + A_y \hat{j}$

For any non-zero vector $\vec{A}$, its direction is isolated as a pure unit vector $\hat{u}_A$ through the ratio of the vector to its magnitude:

$\hat{u}_A = \frac{\vec{A}}{\Vert\vec{A}\Vert}$

Vectors can be added graphically using the head-to-tail method (méthode bout à bout / tête-à-queue), in which the tail of each subsequent vector is placed at the tip of the preceding vector. The resultant vector (vecteur résultant) $\vec{R}$ extends from the tail of the first vector to the tip of the final vector. Vector addition (addition vectorielle) is both commutative (commutative) and associative (associative):

$\vec{A} + \vec{B} = \vec{B} + \vec{A}$

$(\vec{A} + \vec{B}) + \vec{C} = \vec{A} + (\vec{B} + \vec{C})$

All vectors combined in an addition or subtraction operation must share the same physical units and represent the identical physical quantity; adding different physical quantities, such as displacement and velocity, is physically meaningless.

Algebraic vector addition (addition vectorielle algébrique) is executed by summing corresponding components. In two dimensions, given $\vec{R} = \vec{A} + \vec{B}$:

$R_x = A_x + B_x$

$R_y = A_y + B_y$

$\vec{R} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j} = R_x \hat{i} + R_y \hat{j}$

$R = \sqrt{R_x^2 + R_y^2}$

$\theta = \tan^{-1}\left(\frac{R_y}{R_x}\right)$

In three dimensions, the formulation expands to include the $z$-axis:

$\vec{R} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j} + (A_z + B_z)\hat{k} = R_x \hat{i} + R_y \hat{j} + R_z \hat{k}$

$R = \sqrt{R_x^2 + R_y^2 + R_z^2}$

The spatial orientation in three dimensions is determined by directional cosines (cosinus directeurs), such as $\cos(\theta_x) = \frac{R_x}{R}$.

Vector subtraction (soustraction vectorielle) is performed by adding the negative of the vector to be subtracted:

$\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$

The negative vector $-\vec{B}$ maintains the identical magnitude as $\vec{B}$ but points in the directly opposite direction. Graphically, the difference vector $\vec{C} = \vec{A} - \vec{B}$ points from the tip of $\vec{B}$ to the tip of $\vec{A}$.

Multiplying or dividing a vector by a non-zero scalar $s$ alters the vector's magnitude by a factor of $\Vert s\Vert$. If $s$ is positive, the resulting vector maintains the original vector's direction; if $s$ is negative, the resulting vector points in the opposite direction.

Kinematics in One Dimension

Kinematics (cinématique) investigates and describes the motion (mouvement) of objects while disregarding the external agents or forces responsible for producing or modifying that motion. Motion is defined fundamentally as the continual change in an object's position over time. The primary types of motion include translational motion (mouvement de translation) (such as an automobile traveling along a roadway), rotational motion (mouvement de rotation) (such as the rotation of the Earth on its axis), and vibrational motion (mouvement vibratoire) (such as the oscillatory swinging of a pendulum).

To simplify mathematical analysis, physical bodies are frequently approximated using the particle model (modèle particulaire). Under this model, an extended object is treated as a zero-dimensional mathematical point possessing finite mass but infinitesimal physical size.

The position (position) of a particle corresponds to its specific location relative to a chosen spatial reference point, typically defined as the origin (origine) of a coordinate system. In one dimension, motion is restricted to a straight line along a single coordinate axis, such as the $x$-axis.

Displacement (déplacement) is defined as the change in position of an object during a specified time interval:

$\Delta x \equiv x_f - x_i$

Displacement is a vector quantity measured in meters ($\text{m}$) in SI units, capable of taking on positive, negative, or zero values depending on whether motion proceeds in the positive or negative coordinate direction. Displacement must be strictly distinguished from distance (distance). Distance is a scalar quantity equal to the total length of the continuous path traversed by the particle, and it is always positive or zero. For example, if an athlete runs from one baseline of a basketball court to the opposite end and returns back to the starting baseline, the total distance traveled is equal to twice the length of the court, whereas the net displacement is zero because $x_f = x_i$.

Average velocity (vitesse moyenne) is the rate at which displacement occurs over a finite time interval $\Delta t = t_f - t_i$:

$v_{x,\text{avg}} \equiv \frac{\Delta x}{\Delta t} = \frac{x_f - x_i}{t_f - t_i}$

Average velocity possesses dimensions of length divided by time ($[L/T]$) with SI units of $\text{m/s}$. Graphically, the average velocity between two points on a position-time ($x\text{-}t$) curve represents the slope of the secant line (sécante) connecting those points.

Average speed (vitesse scalaire moyenne) is a scalar quantity defined as the total distance traveled $d$ divided by the total elapsed time interval $\Delta t$:

$\text{Average Speed} \equiv \frac{d}{\Delta t}$

Average speed is non-directional, has units of $\text{m/s}$, and is always positive. Average speed is not generally equal to the magnitude of the average velocity. For an object executing a round trip where initial and final positions coincide, the net displacement is zero, yielding an average velocity of $0\,\text{m/s}$, while the distance traveled is non-zero, resulting in a strictly positive average speed.

Instantaneous velocity (vitesse instantanée) is defined as the limit of the average velocity as the elapsed time interval approaches zero, equivalent to the first derivative (première dérivée) of position with respect to time:

$v_x = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt}$

Instantaneous velocity reflects the state of motion at every single instant and corresponds geometrically to the slope of the tangent line (tangente) to the position-time curve at that specific point in time. The instantaneous speed (vitesse instantanée scalaire) is the absolute magnitude of the instantaneous velocity: $v = \Vert v_x\Vert$.

A particle moving under constant velocity (vitesse constante) exhibits identical instantaneous and average velocities across any time interval ($v_x = v_{x,\text{avg}}$). Taking the initial time as $t_i = 0$, its kinematic position equation is:

$x_f = x_i + v_x t$

The position-time graph of a particle under constant velocity is a straight line where the slope equals $v_x$ and the vertical intercept corresponds to $x_i$. A particle moving with constant speed along a curved path conforms to the same rate equation, where constant speed is given by $v = \frac{d}{\Delta t}$.

Acceleration (accélération) represents the time rate of change of velocity. The average acceleration (accélération moyenne) over a time interval $\Delta t$ is defined as:

$a_{x,\text{avg}} \equiv \frac{\Delta v_x}{\Delta t} = \frac{v_{xf} - v_{xi}}{t_f - t_i}$

Instantaneous acceleration (accélération instantanée) is the limiting value of the average acceleration as $\Delta t$ approaches zero, corresponding to the derivative of velocity with respect to time and the second derivative (seconde dérivée) of position with respect to time:

$a_x = \lim_{\Delta t \to 0} \frac{\Delta v_x}{\Delta t} = \frac{dv_x}{dt} = \frac{d^2x}{dt^2}$

Acceleration has dimensions of $[L/T^2]$ and SI units of $\text{m/s}^2$. On a velocity-time graph, instantaneous acceleration corresponds to the slope of the line tangent to the curve at a particular instant.

Acceleration is directly related to the net external force exerted on the object ($F_x \propto a_x$). When velocity and acceleration share the same algebraic sign (same direction), the object speeds up (accélère). When velocity and acceleration possess opposite algebraic signs, the object slows down (ralentit). A negative acceleration does not necessarily imply that an object is slowing down; if both the velocity and acceleration are negative, the object is increasing its speed in the negative direction. The colloquial term deceleration (décélération) is avoided in formal kinematics because of its ambiguous meaning.

Motion diagrams (diagrammes de mouvement) illustrate particle movement by superimposing stroboscopic snapshots of an object at equal successive time increments. On these diagrams, red vector arrows represent instantaneous velocities, while purple vector arrows designate accelerations.

For a particle moving in one dimension under constant acceleration (accélération constante), motion is governed by a standard set of kinematic equations (assuming $t_i = 0$):

$v_{xf} = v_{xi} + a_x t$

$v_{x,\text{avg}} = \frac{v_{xi} + v_{xf}}{2}$

$x_f = x_i + \frac{1}{2}(v_{xi} + v_{xf})t$

$x_f = x_i + v_{xi}t + \frac{1}{2}a_x t^2$

$v_{xf}^2 = v_{xi}^2 + 2a_x(x_f - x_i)$

Using calculus (calcul infinitésimal), these kinematic relations are derived by integration (intégration). The displacement equals the definite integral (intégrale définie) of velocity over time, representing the area beneath the curve on a velocity-time plot:

$\lim_{\Delta t \to 0} \sum_n v_{xn}(t)\Delta t = \int_{t_i}^{t_f} v_x(t)\,dt = x_f - x_i$

Integrating constant acceleration yields:

$\int_{v_{xi}}^{v_{xf}} dv_x = \int_0^t a_x\,dt \implies v_{xf} - v_{xi} = a_x t$

$\int_{x_i}^{x_f} dx = \int_0^t (v_{xi} + a_x t)\,dt \implies x_f - x_i = v_{xi}t + \frac{1}{2}a_x t^2$

Free Fall and Vertical Motion

A freely falling object (objet en chute libre) is any body moving solely under the influence of gravity (gravité), completely independent of its initial state of motion and unaffected by air resistance (résistance de l'air). Regardless of whether an object is dropped from rest, projected downwards, or launched vertically upwards, its acceleration is directed straight down toward the center of the Earth throughout its entire flight.

The magnitude of the acceleration of free fall (accélération de la chute libre) is designated by $g$. At the Earth's surface, the average value is $g = 9.80\,\text{m/s}^2$. The exact value of $g$ varies slightly with latitude (latitude) and decreases with increasing altitude (altitude) above sea level. Defining the vertically upward direction as positive, the vertical acceleration in the kinematic equations is set to $a_y = -g = -9.80\,\text{m/s}^2$.

In the case of an object dropped from rest, the initial velocity is $v_{yi} = 0\,\text{m/s}$ and the acceleration is $a_y = -9.80\,\text{m/s}^2$. When an object is thrown downward, its initial velocity is non-zero and negative ($v_{yi} < 0$) under an upward-positive coordinate convention, accompanied by $a_y = -9.80\,\text{m/s}^2$.

When an object is thrown vertically upward, its initial velocity is positive ($v_{yi} > 0$). Throughout the upward flight, the acceleration remains continuously directed downward at $a_y = -9.80\,\text{m/s}^2$. At the trajectory's maximum peak height, the instantaneous vertical velocity momentarily becomes zero ($v_y = 0\,\text{m/s}$), although the acceleration remains unchanged at $-9.80\,\text{m/s}^2$. For symmetrical vertical flights returning to the original release height, the time of ascent equals the time of descent ($t_{\text{up}} = t_{\text{down}}$), and the return velocity has a magnitude equal to the initial velocity but points downward ($v = -v_0$). When the motion is asymmetric, such as an object launched from an elevated point and landing below its initial height, the trajectory is analyzed by decomposing the motion into distinct ascending and descending segments, tracking displacement relative to the initial origin ($y_i = 0$).

Kinematics in Two Dimensions and Projectile Motion

Two-dimensional motion (mouvement bidimensionnel) requires full vector notation, as algebraic positive and negative signs are insufficient to describe arbitrary directions in a plane. The position of a particle in two dimensions is defined by its position vector (vecteur position) $\vec{r}(t) = x(t)\hat{i} + y(t)\hat{j}$. Displacement across a time interval is given by:

$\Delta\vec{r} = \vec{r}_f - \vec{r}_i$

Average velocity (vitesse moyenne) is the vector displacement divided by the elapsed time:

$\vec{v}_{\text{avg}} = \frac{\Delta\vec{r}}{\Delta t}$

The direction of the average velocity vector aligns precisely with the direction of $\Delta\vec{r}$, remaining completely independent of the actual path traversed between the endpoints. Instantaneous velocity (vitesse instantanée) is the time derivative of the position vector:

$\vec{v} = \lim_{\Delta t \to 0} \frac{\Delta\vec{r}}{\Delta t} = \frac{d\vec{r}}{dt}$

The instantaneous velocity vector is always tangent to the trajectory path (trajectoire) at any given point and points in the direction of motion. Its magnitude represents the instantaneous speed.

Average acceleration (accélération moyenne) is the change in instantaneous velocity divided by the time interval:

$\vec{a}_{\text{avg}} = \frac{\Delta\vec{v}}{\Delta t} = \frac{\vec{v}_f - \vec{v}_i}{t_f - t_i}$

Instantaneous acceleration (accélération instantanée) is the derivative of the velocity vector with respect to time:

$\vec{a} = \lim_{\Delta t \to 0} \frac{\Delta\vec{v}}{\Delta t} = \frac{d\vec{v}}{dt} = \frac{d^2\vec{r}}{dt^2}$

A body experiences non-zero acceleration whenever its velocity vector changes in magnitude, changes in direction, or changes in both simultaneously.

Two-dimensional motion under constant acceleration can be modeled as two independent, decoupled one-dimensional motions along perpendicular axes ($x$ and $y$). Influences acting exclusively along the $y$-direction have no effect on motion along the $x$-direction, and vice versa. The general vector kinematic equations for constant acceleration are:

$\vec{v}_f = \vec{v}_i + \vec{a}t$

$\vec{r}_f = \vec{r}_i + \vec{v}_i t + \frac{1}{2}\vec{a}t^2$

Decomposed into Cartesian component equations, these relationships become:

$v_{xf} = v_{xi} + a_x t$

$v_{yf} = v_{yi} + a_y t$

$x_f = x_i + v_{xi}t + \frac{1}{2}a_x t^2$

$y_f = y_i + v_{yi}t + \frac{1}{2}a_y t^2$

Consider an example in which a particle moves in the $xy$-plane, starting from the origin ($x_i = 0$, $y_i = 0$) at $t = 0$ with an initial velocity vector having components $v_{xi} = 20\,\text{m/s}$ and $v_{yi} = -15\,\text{m/s}$. The particle experiences an acceleration along the $x$-direction of $a_x = 4.0\,\text{m/s}^2$, while $a_y = 0\,\text{m/s}^2$.

The total velocity vector as a function of time is given by:

$\vec{v}(t) = (v_{xi} + a_x t)\hat{i} + v_{yi}\hat{j} = (20 + 4.0t)\hat{i} - 15\hat{j}\,\text{m/s}$

At $t = 5.0\,\text{s}$, the velocity components evaluate to $v_x = 20 + 4.0(5.0) = 40\,\text{m/s}$ and $v_y = -15\,\text{m/s}$. The resulting speed is:

$v = \sqrt{v_x^2 + v_y^2} = \sqrt{40^2 + (-15)^2} = \sqrt{1600 + 225} = \sqrt{1825} \approx 42.7\,\text{m/s}$

The angle that the velocity vector makes with the positive $x$-axis is:

$\theta = \tan^{-1}\left(\frac{v_y}{v_x}\right) = \tan^{-1}\left(\frac{-15}{40}\right) \approx -20.6^\circ$

The particle's position coordinates and position vector at any elapsed time $t$ are:

$x(t) = v_{xi}t + \frac{1}{2}a_x t^2 = 20t + 2.0t^2$

$y(t) = v_{yi}t = -15t$

$\vec{r}(t) = (20t + 2.0t^2)\hat{i} - 15t\hat{j}$

Projectile motion (mouvement de projectile) represents a classic form of two-dimensional motion where an object is launched into the air near the Earth's surface and moves freely under gravitational acceleration. The standard physical analysis relies on two fundamental assumptions: air resistance (résistance de l'air) is negligible, and the gravitational acceleration vector is constant, points downward, and has magnitude $g = 9.80\,\text{m/s}^2$. This assumption holds provided the projectile's horizontal range remains small compared to the radius of the Earth. Under these conditions, the path of the projectile is a parabola (parabole), referred to as its trajectory (trajectoire).

Projectile motion is resolved into the superposition of two independent linear motions: uniform velocity motion along the horizontal axis ($a_x = 0$), and uniform acceleration free fall along the vertical axis ($a_y = -g = -9.80\,\text{m/s}^2$). Resolving initial launch velocity $v_i$ at angle $\theta$ above the horizontal yields:

$v_{xi} = v_i \cos(\theta)$

$v_{yi} = v_i \sin(\theta)$

The kinematic equations governing projectile motion across time $t$ are:

$v_x(t) = v_i \cos(\theta)$

$x(t) = v_i \cos(\theta)t$

$v_y(t) = v_i \sin(\theta) - gt$

$y(t) = v_i \sin(\theta)t - \frac{1}{2}gt^2$

The total horizontal distance covered across the complete flight is designated as the horizontal range (portée horizontale) $R$, while the peak vertical displacement represents the maximum height (hauteur maximale) $h$. The maximum horizontal range over level ground occurs at a launch angle of $\theta_i = 45^\circ$. Launch angles that are complementary (angles complémentaires) ($\theta_1 + \theta_2 = 90^\circ$) yield identical horizontal ranges, although their maximum heights and total flight times differ.

Analyzing complex physical problems follows a structured four-stage problem-solving methodology: Conceptualize (conceptualiser), Categorize (catégoriser), Analyze (analyser), and Finalize (finaliser). In the Conceptualize step, the physical scenario is visualized, represented through sketches, and parsed for stated parameters and units. In the Categorize step, simplifications are established (such as modeling extended bodies as point particles and omitting negligible friction), and the appropriate theoretical model is identified (such as a particle under constant velocity or constant acceleration). In the Analyze step, the relevant equations are identified, solved symbolically for the unknown quantities, and evaluated with appropriate units and significant figures. In the Finalize step, numerical answers are checked for physical realism, proper units, consistency with limiting cases, and agreement with original qualitative expectations.

Dynamics and Newton's Laws of Motion

While kinematics describes motion, dynamics (dynamique) examines the physical causes of motion, identifying how forces produce, alter, or maintain mechanical motion, as well as the conditions necessary for a body to maintain static or dynamic equilibrium (équilibre statique ou dynamique). The classical foundation of dynamics was established by Sir Isaac Newton (1642–1727), who formulated the basic laws of mechanics, discovered the law of universal gravitation (loi de la gravitation universelle), developed mathematical calculus (calcul infinitésimal), and performed pioneering investigations into optics (optique).

A force (force) is broadly defined as any physical interaction that produces an acceleration (accélération) (a change in the state of motion) or causes the mechanical deformation (déformation mécanique) of an object. In Newtonian mechanics, a force is defined strictly as that which causes acceleration. All forces are categorized as either a push (poussée) or a pull (traction).

In classical mechanics, forces are categorized into contact forces (forces de contact) and field forces (forces de champ):

Contact forces involve direct physical contact between macroscopic objects. Examples include compressing or extending a mechanical spring, pulling a wagon via a handle, or kicking a football. Field forces act across empty space without requiring physical contact. Examples include the gravitational force (force gravitationnelle) between masses, the electrostatic force (force électrostatique) between electrical charges, and the magnetic force (force magnétique) exerted by a magnet on iron.

Forces are vector quantities possessing both magnitude and direction, obeying all standard rules of vector algebra. The net force (force nette / résultante) (also designated as the resultant force, total force, or unbalanced force) acting on a body is the vector sum of all individual forces acting upon that body:

$\vec{F}_{\text{net}} = \sum \vec{F}$

Newton's First Law (première loi de Newton) states that in the absence of external forces, an object at rest remains at rest, and an object in motion continues in motion with a constant velocity (constant speed in a straight line). Newton's First Law describes the natural behavior of an isolated object in the absence of net external forces, establishing that when no external force acts on an object, its acceleration is zero. An isolated object naturally maintains a constant velocity (which includes remaining at rest). The first law operationalizes the concept of force as the external influence required to change an object's state of motion.

Inertia (inertie) is the inherent tendency of an object to resist any attempt to alter its state of rest or uniform motion. Inertia represents a fundamental qualitative physical concept rather than a directly measurable quantitative property. Mass (masse), by contrast, is a true physical scalar quantity that provides a quantitative measure of an object's inertia. Mass reflects the quantity of matter contained within an object, remains completely independent of the object's external surroundings, and is measured in kilograms ($\text{kg}$).

Mass must not be confused with weight (poids). Weight is a gravitational force exerted upon an object by a celestial body, such as the Earth. The weight force always points directly toward the center of the gravitating planet, and its magnitude is given by:

$W = mg$

Because the local gravitational acceleration $g$ varies across spatial locations and decreases at higher altitudes, an object's weight varies depending on location, whereas its mass remains strictly invariant. For example, an astronaut with a mass of $50\,\text{kg}$ has a weight of $W = (50\,\text{kg})(9.8\,\text{m/s}^2) = 490\,\text{N}$ on Earth. On Mars, where gravitational acceleration is lower, the astronaut's mass remains $50\,\text{kg}$, but their weight decreases to $186\,\text{N}$.

Newton's Second Law (deuxième loi de Newton) states that the acceleration of an object is directly proportional to the net resultant force acting upon it and inversely proportional to its mass:

$\vec{a} \propto \frac{\sum \vec{F}}{m}$

Setting the constant of proportionality to 1 defines the equation of motion:

$\sum \vec{F} = m\vec{a}$

Newton's Second Law is valid for speeds substantially lower than the speed of light ($v \ll c$). In three-dimensional Cartesian coordinates, this vector relationship decouples into three independent scalar component equations:

$\sum F_x = m a_x$

$\sum F_y = m a_y$

$\sum F_z = m a_z$

The SI unit of force is the newton ($\text{N}$) (newton). One newton is defined as the net force required to accelerate a mass of one kilogram at a rate of one meter per second squared:

$1\,\text{N} \equiv 1\,\text{kg}\cdot\text{m/s}^2$

Newton's Third Law (troisième loi de Newton) states that if two objects interact, the force $\vec{F}_{12}$ exerted by object 1 on object 2 is equal in magnitude and opposite in direction to the force $\vec{F}_{21}$ exerted by object 2 on object 1:

$\vec{F}_{12} = -\vec{F}_{21}$

This principle is known as the action-reaction law (loi de l'action-réaction). The action force (force d'action) and reaction force (force de réaction) occur simultaneously, share the identical physical nature, and act on different objects. Because the two forces act on entirely separate bodies, they never cancel each other out when analyzing the dynamics of a single body.

Specific Types of Forces and Free-Body Diagrams

The normal force (force normale) ($\vec{n}$) is the contact reaction force exerted by a supporting surface on an object resting against it. The term normal signifies that this force acts perpendicular (perpendiculaire) to the contact interface, counteracting the forces pressing the body against the surface. The normal force is not an inherent property of the body and is not universally equal to its weight $mg$. For a mass resting unconstrained on a horizontal plane, static vertical equilibrium gives $n = mg$. If an external downward vertical force $\vec{F}$ presses down on the object, the normal force increases to support both the weight and the applied load:

$\vec{n} = \vec{F} + \vec{F}_g$

$n = F + mg$

Tension (tension) ($\vec{T}$) is the pull force transmitted along a continuous flexible medium, such as a taut rope, cable, or string. The tension force always acts along the axis of the line and points directly away from the object connected to that line.

Friction (frottement) ($\vec{f}$) is a contact resistance force that opposes the relative motion or impending sliding of an object across a rough surface or through a viscous medium. The friction force acts parallel (parallèle) to the contact surface and points opposite to the direction of motion or intended motion.

A free-body diagram (FBD) (diagramme du corps libre) is a graphical representation used to isolate an object and analyze the forces acting upon it. In a free-body diagram, the object is modeled as a single dot (the particle model (modèle particulaire)). All external forces acting directly upon that body are drawn as directed vector arrows pointing away from the dot. Reaction forces exerted by the object onto other external bodies are strictly excluded.

For a crate pulled across a rough horizontal surface by a rope, the free-body diagram isolates four concurrent external forces: the upward normal force $\vec{n}$, the downward gravitational weight force $\vec{F}_g$, the forward horizontal tension force $\vec{T}$, and the backward horizontal friction force $\vec{f}$.

Dynamic Analysis of Vertical Circular Motion

Newton's second law governs objects experiencing centripetal acceleration (accélération centripète) during curved or circular motion (mouvement circulaire). Consider an automobile traveling at a constant speed $v$ over a circular hill of radius $R$ and subsequently through a circular valley possessing the identical radius of curvature $R$.

Let the driver have a mass of $m = 70.0\,\text{kg}$. At the crest of the circular hill, the driver moves along an arc whose center of curvature lies vertically below the car. The forces acting on the driver along the vertical axis are the downward gravitational force $mg$ and the upward normal contact force $n_{\text{top}}$ exerted by the seat. The downward centripetal acceleration is $a_c = \frac{v^2}{R}$. Applying Newton's second law along the inward radial direction gives:

$\sum F_r = mg - n_{\text{top}} = m\left(\frac{v^2}{R}\right)$

If the car passes over the hill at a speed such that the driver experiences apparent weightlessness (apesanteur apparente), the normal force from the seat vanishes ($n_{\text{top}} = 0$). Substituting this boundary condition into the equation of motion determines the centripetal term:

$mg - 0 = m\left(\frac{v^2}{R}\right) \implies \frac{mv^2}{R} = mg$

When the vehicle maintains the identical constant speed $v$ while traversing the bottom of the circular valley of radius $R$, the center of curvature lies vertically above the car. The vertical forces acting on the driver remain the downward gravitational force $mg$ and an upward normal force $n_{\text{bottom}}$ exerted by the seat. The centripetal acceleration now points vertically upward toward the center of curvature:

$\sum F_r = n_{\text{bottom}} - mg = m\left(\frac{v^2}{R}\right)$

Substituting the relationship $\frac{mv^2}{R} = mg$ into this dynamic balance yields:

$n_{\text{bottom}} - mg = mg$

$n_{\text{bottom}} = 2mg$

Evaluating this result with the driver's mass of $m = 70.0\,\text{kg}$ and standard gravitational acceleration $g = 9.80\,\text{m/s}^2$:

$n_{\text{bottom}} = 2(70.0\,\text{kg})(9.80\,\text{m/s}^2) = 1372\,\text{N}$

The normal force exerted by the seat on the driver at the valley floor is twice the driver's normal resting weight, producing an apparent weight (poids apparent) of $1372\,\text{N}$.