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

Una comprobación de seguridad en un barco utiliza un peso de prueba pesado. El peso se levanta y luego se libera para estudiar cómo la resistencia del aire afecta al movimiento. Una vez que se suelta, el peso parte del reposo y cae por el aire.La gravedad lo tira hacia abajo, mientras el aire empuja hacia arriba contra su movimiento. Según la Segunda Ley de Newton, el cambio en la velocidad depende de la fuerza neta.Combinando estas fuerzas, se obtiene una ecuación diferencial que vincula la aceleración con la velocidad. Dividir la ecuación por la masa da una forma más sencilla.Definir la relación entre la constante de resistencia y la masa como la constante b facilita la separación de la ecuación diferencial.Integrar la ecuación y reescribirla para encontrar la ecuación de la velocidad en función del tiempo da como resultado exponencial. Usar la velocidad inicial cero ayuda a encontrar la constante restante en la solución.A medida que el tiempo avanza, la velocidad se acerca a un valor constante conocido como velocidad terminal. Con un peso de 10 kilogramos y una constante de resistencia de 2 newton-segundo por metro, el modelo predice una velocidad terminal de 49 metros por segundo.

ecuaciones diferenciales: resolución de problemas

Al analizar el movimiento de objetos que caen, es fundamental considerar no solo la fuerza de la gravedad, sino tambi&#xE9;n la fuerza opuesta debida a la resistencia del aire. Un ejemplo pr&#xE1;ctico consiste en dejar caer un peso de prueba pesado durante una comprobaci&#xF3;n de seguridad en un barco. Al caer desde el reposo, la gravedad lo acelera hacia abajo, mientras que la resistencia del aire ejerce una fuerza ascendente que aumenta con la velocidad. Esta interacci&#xF3;n din&#xE1;mica de fuerzas se describe bien mediante ecuaciones diferenciales, que proporcionan un marco matem&#xE1;tico para modelar la velocidad cambiante del objeto a lo largo del tiempo.
Fuerzas y modelado diferencial
Seg&#xFA;n la segunda ley de Newton, la fuerza neta sobre el peso que cae determina su aceleraci&#xF3;n. La gravedad ejerce una fuerza constante igual a la masa del objeto multiplicada por la aceleraci&#xF3;n de la gravedad, mientras que la resistencia del aire se modela habitualmente como proporcional a la velocidad del objeto. Al combinar estas fuerzas, la fuerza neta produce una ecuaci&#xF3;n diferencial de primer orden que relaciona la tasa de cambio de la velocidad con la propia velocidad.
Comportamiento exponencial y velocidad terminal
Al resolver la ecuaci&#xF3;n diferencial resultante, se obtiene una funci&#xF3;n de velocidad que aumenta con el tiempo, pero que se aproxima asint&#xF3;ticamente a un l&#xED;mite finito. Este comportamiento refleja el equilibrio gradual entre la gravedad y la resistencia del aire, que culmina en un estado conocido como velocidad terminal: el punto en el que cesa la aceleraci&#xF3;n y el objeto cae a velocidad constante. Con una masa de 10 kg y una constante de arrastre de 2 N&#xB7;s/m, la velocidad terminal calculada es de 49 m/s. Este resultado ilustra c&#xF3;mo las ecuaciones diferenciales modelan eficazmente el movimiento en situaciones reales y revelan el papel de la resistencia del aire en la limitaci&#xF3;n de la aceleraci&#xF3;n durante la ca&#xED;da libre.

Ecuaciones diferenciales: resolución de problemas

Al analizar el movimiento de objetos que caen, es fundamental considerar no solo la fuerza de la gravedad, sino tambi&#xE9;n la fuerza opuesta debida a la ...

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Differential Equations

Ecuaciones diferenciales

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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