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

Bir gemide yapılan güvenlik kontrolünde ağır bir test ağırlığı kullanılır. Ağırlık kaldırılır ve ardından hava direncinin hareketi nasıl etkilediğini incelemek için serbest bırakılır. Bırakıldığında, ağırlık dinlenmeden başlar ve havaya düşer.Yerçekimi onu aşağı çekerken, hava hareketine karşı yukarıya doğru iter. Newton'un İkinci Yasası'na göre, hız değişimi net kuvvete bağlıdır.Bu kuvvetlerin birleştirilmesi, ivmeyi hızla ilişkilendiren bir diferansiyel denklem ortaya çıkar. Denklemi kütle bölmek daha basit bir form verir.Sürtünme sabitinin kütle ile oranını b sabiti olarak tanımlamak, diferansiyel denklemi ayırmayı kolaylaştırır.Denklemi entegre edip yeniden yazarak hız denklemini zaman fonksiyonu olarak bulursak, üstel bir denklem elde edilir. Sıfırın başlangıç hızını kullanmak, çözeltide kalan sabiti bulmaya yardımcı olur.Zaman arttıkça, hız terminal hız olarak bilinen sabit bir değere yaklaşır. 10 kilogram ağırlık ve metre başına 2 newton-saniye sürtünme sabitiyle model, saniyede 49 metre terminal hız öngörüyor.

diferansiyel denklemler: problem Çözümü

D&#xFC;&#x15F;en cisimlerin hareketini analiz ederken, yaln&#x131;zca yer&#xE7;ekimi kuvvetini de&#x11F;il, ayn&#x131; zamanda d&#xFC;&#x15F;&#xFC;&#x15F; h&#x131;z&#x131;na kar&#x15F;&#x131; koyan hava direnci kuvvetini de dikkate almak gerekir. Uygulamal&#x131; bir &#xF6;rnek olarak, bir gemide g&#xFC;venlik kontrol&#xFC; s&#x131;ras&#x131;nda a&#x11F;&#x131;r bir test y&#xFC;k&#xFC;n&#xFC;n serbest b&#x131;rak&#x131;lmas&#x131; verilebilir. Y&#xFC;k durgun h&#xE2;lde iken serbest b&#x131;rak&#x131;l&#x131;p d&#xFC;&#x15F;erken, yer&#xE7;ekimi onu a&#x15F;a&#x11F;&#x131; y&#xF6;nde ivmelendirir; hava direnci ise h&#x131;zla birlikte artan ve yukar&#x131; y&#xF6;nde etki eden bir kuvvet uygular. Kuvvetler aras&#x131;ndaki bu dinamik etkile&#x15F;im, cismin zaman i&#xE7;inde de&#x11F;i&#x15F;en h&#x131;z&#x131;n&#x131; modellemek i&#xE7;in matematiksel bir &#xE7;er&#xE7;eve sunan diferansiyel denklemlerle iyi bi&#xE7;imde modellenir.
Kuvvetler ve Diferansiyel Modelleme
Newton&#x2019;un &#x130;kinci Yasas&#x131;&#x2019;na g&#xF6;re, d&#xFC;&#x15F;en y&#xFC;k &#xFC;zerindeki net kuvvet, bu y&#xFC;k&#xFC;n ivmesini belirler. Yer&#xE7;ekimi kuvveti, cismin k&#xFC;tlesi ile yer&#xE7;ekimi ivmesinin &#xE7;arp&#x131;m&#x131;na e&#x15F;it ve sabittir; hava direnci ise genellikle cismin h&#x131;z&#x131;yla orant&#x131;l&#x131; olarak modellenir. Bu kuvvetler bir araya getirildi&#x11F;inde, net kuvvet, h&#x131;z&#x131;n zamana g&#xF6;re de&#x11F;i&#x15F;im oran&#x131;n&#x131; h&#x131;z ile ili&#x15F;kilendiren birinci mertebeden bir diferansiyel denkleme kar&#x15F;&#x131;l&#x131;k gelir.
&#xDC;stel Davran&#x131;&#x15F; ve Son H&#x131;z
Ortaya &#xE7;&#x131;kan diferansiyel denklemin &#xE7;&#xF6;z&#xFC;m&#xFC;, zamanla artan ancak asimptotik olarak sonlu bir s&#x131;n&#x131;ra yakla&#x15F;an bir h&#x131;z fonksiyonu verir. Bu davran&#x131;&#x15F;, yer&#xE7;ekimi ile hava direncinin giderek dengelenmesini yans&#x131;t&#x131;r ve ivmenin s&#x131;f&#x131;ra indi&#x11F;i, cismin sabit bir h&#x131;zla d&#xFC;&#x15F;t&#xFC;&#x11F;&#xFC; son h&#x131;z durumuyla sonu&#xE7;lan&#x131;r. K&#xFC;tlenin 10 kg ve s&#xFC;r&#xFC;kleme katsay&#x131;s&#x131;n&#x131;n 2 N&#xB7;s/m oldu&#x11F;u durumda hesaplanan son h&#x131;z 49 m/s&#x2019;dir. Bu sonu&#xE7;, diferansiyel denklemlerin ger&#xE7;ek d&#xFC;nya hareketini etkili bi&#xE7;imde modelledi&#x11F;ini ve serbest d&#xFC;&#x15F;&#xFC;&#x15F; s&#x131;ras&#x131;nda ivmeyi s&#x131;n&#x131;rlamada hava direncinin rol&#xFC;n&#xFC; ortaya koydu&#x11F;unu g&#xF6;stermektedir.

Diferansiyel Denklemler: Problem Çözümü

D&#xFC;&#x15F;en cisimlerin hareketini analiz ederken, yaln&#x131;zca yer&#xE7;ekimi kuvvetini de&#x11F;il, ayn&#x131; zamanda d&#xFC;&#x15F;&#xFC;&#x15F; ...

Education

JoVE Core

Calculus

Kalkülüs

Unit

Birim

Differential Equations

Diferansiyel Denklemler

KEY TERMS AND CONCEPTS

TEMEL TERİMLER VE KAVRAMLAR

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

Description

Transcript

Tags