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

Uma verificação de segurança em um navio utiliza um peso de teste pesado. O peso é levantado e depois liberado para estudar como a resistência do ar afeta o movimento. Depois de solta, o peso começa do repouso e cai pelo ar.A gravidade o puxa para baixo, enquanto o ar empurra para cima contra seu movimento. De acordo com a Segunda Lei de Newton, a mudança na velocidade depende da força líquida.A combinação dessas forças gera uma equação diferencial que liga aceleração à velocidade. Dividir a equação pela massa dá uma forma mais simples.Definir a razão entre a constante de arrasto e a massa como a constante b facilita a separação da equação diferencial.Integrar a equação e reescrevê-la para encontrar a equação da velocidade em função do tempo obtém uma equação exponencial. Usar a velocidade inicial zero ajuda a encontrar a constante restante na solução.À medida que o tempo avança, a velocidade se aproxima de um valor constante conhecido como velocidade terminal. Com peso de 10 quilos e constante de arrasto de 2 newtons-segundo por metro, o modelo prevê uma velocidade terminal de 49 metros por segundo.

equações diferenciais: resolução de problemas

Ao analisar o movimento de objetos em queda, &#xE9; essencial considerar n&#xE3;o apenas a for&#xE7;a da gravidade, mas tamb&#xE9;m a resist&#xEA;ncia do ar, que se op&#xF5;e ao movimento. Um exemplo pr&#xE1;tico envolve a libera&#xE7;&#xE3;o de um peso de teste pesado durante uma inspe&#xE7;&#xE3;o de seguran&#xE7;a em um navio. &#xC0; medida que o peso cai a partir do repouso, a gravidade o acelera para baixo, enquanto a resist&#xEA;ncia do ar exerce uma for&#xE7;a para cima que aumenta com a velocidade. Essa intera&#xE7;&#xE3;o din&#xE2;mica de for&#xE7;as &#xE9; adequadamente descrita por equa&#xE7;&#xF5;es diferenciais, que fornecem uma estrutura matem&#xE1;tica para modelar a varia&#xE7;&#xE3;o da velocidade do objeto ao longo do tempo.
For&#xE7;as e Modelagem Diferencial
De acordo com a Segunda Lei de Newton, a for&#xE7;a resultante sobre o peso em queda determina sua acelera&#xE7;&#xE3;o. A gravidade exerce uma for&#xE7;a constante igual &#xE0; massa do objeto multiplicada pela acelera&#xE7;&#xE3;o gravitacional, enquanto a resist&#xEA;ncia do ar &#xE9; geralmente modelada como proporcional &#xE0; velocidade do objeto. Quando essas for&#xE7;as s&#xE3;o combinadas, a for&#xE7;a resultante resulta em uma equa&#xE7;&#xE3;o diferencial de primeira ordem que relaciona a taxa de varia&#xE7;&#xE3;o da velocidade com a pr&#xF3;pria velocidade.
Comportamento Exponencial e Velocidade Terminal
Resolver a equa&#xE7;&#xE3;o diferencial resultante fornece uma fun&#xE7;&#xE3;o de velocidade que aumenta com o tempo, mas se aproxima assintoticamente de um limite finito. Esse comportamento reflete o equil&#xED;brio gradual entre a gravidade e a resist&#xEA;ncia do ar, culminando em um estado conhecido como velocidade terminal &#x2014; o ponto em que a acelera&#xE7;&#xE3;o cessa e o objeto cai com velocidade constante. Com uma massa de 10 kg e uma constante de arrasto de 2 N&#xB7;s/m, a velocidade terminal calculada &#xE9; 49 m/s. Esse resultado ilustra como as equa&#xE7;&#xF5;es diferenciais modelam com efic&#xE1;cia o movimento em situa&#xE7;&#xF5;es reais e revelam o papel da resist&#xEA;ncia do ar na limita&#xE7;&#xE3;o da acelera&#xE7;&#xE3;o durante a queda livre.

Equações Diferenciais: Resolução de Problemas

Ao analisar o movimento de objetos em queda, &#xE9; essencial considerar n&#xE3;o apenas a for&#xE7;a da gravidade, mas tamb&#xE9;m a resist&#xEA;ncia do ...

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Calculus

Cálculo

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Differential Equations

Equações Diferenciais

KEY TERMS AND CONCEPTS

TERMOS-CHAVE E CONCEITOS

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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