Name: differential equations and falling motion
Uploaded: 2026-01-12T12:30:58.000Z
Duration: 1 min 21 s
Channel: JoVE

Un contrôle de sécurité sur un navire utilise un poids d’essai lourd. Le poids est soulevé puis relâché pour étudier comment la résistance de l’air affecte le mouvement. Une fois relâché, le poids commence par le repos et tombe dans l’air.La gravité la tire vers le bas, tandis que l’air pousse vers le haut contre son mouvement. Selon la deuxième loi de Newton, la variation de vitesse dépend de la force nette.La combinaison de ces forces donne une équation différentielle qui relie l’accélération à la vitesse. Diviser l’équation par la masse donne une forme plus simple.Définir le rapport de la constante de traînée à la masse comme la constante b facilite la séparation de l’équation différentielle.En intégrant l’équation et en la réécrivant pour trouver l’équation de la vitesse en fonction du temps, on obtient une équation exponentielle. Utiliser la vitesse initiale zéro aide à trouver la constante restante dans la solution.Au fil du temps, la vitesse tend vers une valeur constante appelée vitesse terminale. Avec un poids de 10 kilogrammes et une constante de traînée de 2 newtons-secondes par mètre, le modèle prédit une vitesse terminale de 49 mètres par seconde.

Équations différentielles : résolution de problèmes

Lors de l&#x2019;analyse du mouvement d&#x2019;objets en chute libre, il est essentiel de prendre en compte non seulement la force de gravit&#xE9;, mais aussi la force de r&#xE9;sistance de l&#x2019;air. Un exemple concret consiste &#xE0; laisser tomber une masse d&#x2019;essai lourde lors d&#x2019;un contr&#xF4;le de s&#xE9;curit&#xE9; sur un navire. Alors que la masse tombe &#xE0; partir du repos, la gravit&#xE9; l&#x2019;acc&#xE9;l&#xE8;re vers le bas tandis que la r&#xE9;sistance de l&#x2019;air exerce une force ascendante qui augmente avec la vitesse. Cette interaction dynamique des forces est bien d&#xE9;crite par des &#xE9;quations diff&#xE9;rentielles, qui fournissent un cadre math&#xE9;matique pour mod&#xE9;liser l&#x2019;&#xE9;volution de la vitesse de l&#x2019;objet au fil du temps.
Forces et mod&#xE9;lisation diff&#xE9;rentielle
Selon la deuxi&#xE8;me loi de Newton, la force nette agissant sur le poids en chute libre d&#xE9;termine son acc&#xE9;l&#xE9;ration. La gravit&#xE9; exerce une force constante &#xE9;gale au produit de la masse de l&#x2019;objet par l&#x2019;acc&#xE9;l&#xE9;ration de la pesanteur, tandis que la r&#xE9;sistance de l&#x2019;air est g&#xE9;n&#xE9;ralement mod&#xE9;lis&#xE9;e comme &#xE9;tant proportionnelle &#xE0; la vitesse de l&#x2019;objet. La combinaison de ces forces donne lieu &#xE0; une &#xE9;quation diff&#xE9;rentielle du premier ordre reliant le taux de variation de la vitesse &#xE0; la vitesse elle-m&#xEA;me.
Comportement exponentiel et vitesse limite
La r&#xE9;solution de l&#x2019;&#xE9;quation diff&#xE9;rentielle obtenue donne une fonction vitesse qui cro&#xEE;t avec le temps mais tend asymptotiquement vers une limite finie. Ce comportement refl&#xE8;te l&#x2019;&#xE9;quilibre progressif entre la gravit&#xE9; et la r&#xE9;sistance de l&#x2019;air, aboutissant &#xE0; un &#xE9;tat appel&#xE9; vitesse limite, c&#x2019;est-&#xE0;-dire le point o&#xF9; l&#x2019;acc&#xE9;l&#xE9;ration cesse et o&#xF9; l&#x2019;objet tombe &#xE0; vitesse constante. Avec une masse de 10 kg et un coefficient de tra&#xEE;n&#xE9;e de 2 N&#xB7;s/m, la vitesse limite calcul&#xE9;e est 49 m/s. Ce r&#xE9;sultat illustre comment les &#xE9;quations diff&#xE9;rentielles mod&#xE9;lisent efficacement les mouvements r&#xE9;els et r&#xE9;v&#xE8;lent le r&#xF4;le de la r&#xE9;sistance de l&#x2019;air dans la limitation de l&#x2019;acc&#xE9;l&#xE9;ration lors de la chute libre.

Lors de l&#x2019;analyse du mouvement d&#x2019;objets en chute libre, il est essentiel de prendre en compte non seulement la force de gravit&#xE9;, mais ...

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Calculus

Calcul

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Differential Equations

Équations différentielles

KEY TERMS AND CONCEPTS

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Differential Equations: Problem Solving

differential equations and falling motion

Differential equations can model the motion of a falling object when gravity and air resistance act at the same time. In this example, a heavy test weight is released from rest during a ship safety check. Gravity pulls the weight downward, and air resistance pushes upward as the speed increases.

Newton’s Second Law links the net force on the weight to its acceleration. Gravity provides a constant downward force equal to mass times gravitational acceleration. Air resistance is modeled as a force proportional to the object’s velocity, so the upward drag grows as the weight moves faster.

When these forces are combined, the result is a first-order differential equation. It relates how fast the velocity changes to the velocity itself. Solving the equation gives a velocity function that rises over time but levels off toward a finite limit.

That limit is terminal velocity, which is the speed reached when acceleration stops and the object falls at a constant rate. For a 10 kg mass with a drag constant of 2 N·s/m, the terminal velocity is 49 m/s. This example shows how differential equations describe real motion and explain how air resistance limits free fall.

Differential Equations and Falling Motion

differential-equations-and-falling-motion

introduction to differential equations

Modeling Motion with Second-Order Equations

Differential equations connect a function with its derivatives. They are used to describe changing systems in science and engineering. A differential equation is classified by order, which is the highest derivative it contains. A first-order equation includes only the first derivative, while a second-order equation includes the second derivative of the unknown function. A common second-order differential equation comes from a spring-mass system. In this idealized model, a mass is attached to a...

modeling with differential equations

Population Growth: Exponential vs. Logistic

Population growth can be modeled with differential equations by treating population size, P(t), as a function of time. The derivative of P(t) shows how fast the population changes. A simple starting idea is that the growth rate is proportional to the population size. That assumption gives an exponential growth model. In this model, the population rises continuously and faster over time. It can be a useful first approximation, but it does not match long-term behavior in nature. Real...

separable differential equations

Solving Separable Differential Equations

A separable differential equation is a first-order differential equation that can be written with dy/dx as a product of one function of x and one function of y. This form makes it possible to separate the variables so that all y terms stay on one side and all x terms stay on the other. After the variables are separated, each side can be integrated with respect to its own variable. The result is a relationship between x and y. Depending on the functions involved, that relationship may be...

orthogonal trajectories

Perpendicular Curves in Calculus

Orthogonal trajectories are a pair of curve families that cross at right angles. In this example, the starting family is made of parabolas that open sideways along the x-axis. They share the same shape, but a scaling parameter changes how wide each curve is. Every curve in the family passes through the origin and spreads out at a different rate. To find the orthogonal trajectories, the slope of each parabola must be calculated first. The slope of a tangent shows how steep the curve is at a...

linear differential equations

Integrating Factors for Motion with Drag

The integrating factor method solves first-order linear differential equations that are not separable. It is especially useful when a system has a constant driving force and a resistive force that depends on the changing variable. In these problems, the equation can be rearranged so the left-hand side becomes a product rule after multiplication by an integrating factor. A clear example is a car moving under a constant engine force while air resistance acts against its motion. Newton’s second...

differential equations: problem solving

Differential equations can model the motion of a falling object when gravity and air resistance act at the same time. In this example, a heavy test weight is released from rest during a ship safety check. Gravity pulls the weight downward, and air resistance pushes upward as the speed increases. Newton’s Second Law links the net force on the weight to its acceleration. Gravity provides a constant downward force equal to mass times gravitational acceleration. Air resistance is modeled as a...

key terms and concepts

See how differential equations model a falling weight, including gravity, air resistance, and terminal velocity in a ship safety check example.

Differential Equations and Falling Motion | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Differential Equations and Falling Motion

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