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

A safety check on a ship uses a heavy test weight. The weight is lifted and then released to study how air resistance affects motion. Once it is let go, the weight starts from rest and falls through the air.Gravity pulls it downward, while the air pushes upward against its motion. According to Newton&#8217;s Second Law, the change in speed depends on the net force.Combining these forces gives a differential equation that links acceleration to speed. Dividing the equation by mass gives a simpler form.Defining the ratio of the drag constant to the mass as the constant b makes the differential equation easier to separate.Integrating the equation and rewriting it to find the equation for velocity as a function of time gives an exponential equation. Using the initial velocity of zero helps find the remaining constant in the solution.As time increases, the velocity approaches a constant value known as terminal velocity. With a weight of 10 kilograms and a drag constant of 2 newton-seconds per meter, the model predicts a terminal velocity of 49 meters per second.

differentiaalvergelijkingen: probleemoplossen

Bij het analyseren van de beweging van vallende voorwerpen is het essentieel om niet alleen rekening te houden met de zwaartekracht, maar ook met de tegengestelde kracht van de luchtweerstand. Een praktisch voorbeeld betreft het loslaten van een zwaar testgewicht tijdens een veiligheidscontrole op een schip. Terwijl het gewicht vanuit rust valt, versnelt de zwaartekracht het naar beneden, terwijl de luchtweerstand een opwaartse kracht uitoefent die toeneemt met de snelheid. Deze dynamische wisselwerking van krachten wordt goed beschreven door differentiaalvergelijkingen, die een wiskundig kader bieden voor het modelleren van de veranderende snelheid van het object in de tijd.
Krachten en differentiaalmodellering
Volgens de tweede wet van Newton bepaalt de nettokracht op het vallende gewicht de versnelling ervan. De zwaartekracht oefent een constante kracht uit die gelijk is aan de massa van het object vermenigvuldigd met de zwaartekrachtsversnelling, terwijl de luchtweerstand doorgaans wordt gemodelleerd als evenredig met de snelheid van het object. Wanneer deze krachten worden gecombineerd, resulteert de nettokracht in een differentiaalvergelijking van de eerste orde die de verandering van de snelheid in verband brengt met de snelheid zelf.
Exponentieel gedrag en eindsnelheid
Het oplossen van de hieruit voortvloeiende differentiaalvergelijking levert een snelheidsfunctie op die in de loop van de tijd toeneemt, maar asymptotisch een eindige limiet benadert. Dit gedrag weerspiegelt het geleidelijk in evenwicht komen van zwaartekracht en luchtweerstand, wat culmineert in een toestand die bekendstaat als de eindsnelheid &#x2013; het punt waarop de versnelling ophoudt en het object met een constante snelheid valt. Bij een massa van 10 kg en een luchtweerstandsconstante van 2 N&#xB7;s/m bedraagt de berekende eindsnelheid 49 m/s. Dit resultaat illustreert hoe differentiaalvergelijkingen werkelijke beweging effectief modelleren en inzicht bieden in de rol van luchtweerstand bij het beperken van de versnelling tijdens vrije val.

Differentiaalvergelijkingen: probleemoplossen

Bij het analyseren van de beweging van vallende voorwerpen is het essentieel om niet alleen rekening te houden met de zwaartekracht, maar ook met de ...

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Calculus

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Eenheid

Differential Equations

Differentiaalvergelijkingen

KEY TERMS AND CONCEPTS

BELANGRIJKE TERMEN EN CONCEPTEN

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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