Name: quadratic motion and parabolas
Uploaded: 2025-11-26T05:43:59.000Z
Duration: 1 min 23 s
Channel: JoVE

Wenn ein Ball in die Luft geworfen wird, folgt er einer gekrümmten Bahn, die als Projektilbewegung bezeichnet wird – der Weg, den ein Objekt unter dem Einfluss der Schwerkraft nimmt.Diese vertikale Bewegung bei konstanter Beschleunigung wird mit Hilfe einer quadratischen Funktion der Zeit modelliert. Die Funktion besteht aus drei Termen: einem t-Quadrat-Term, einem t-Term und einer Konstante.Der Graph dieser Funktion ist eine nach unten öffnende Parabel. Es zeigt die Kugel, die steigt, ihren höchsten Punkt erreicht und dann fällt.Wenn Sie das Quadrat vervollständigen, wird die Gleichung in die Standardform einer Parabel umgeschrieben, wodurch der Scheitelpunkt – der höchste Punkt der Bewegung – sichtbar wird. Der Zeitpunkt, zu dem dies geschieht, ist durch negatives b über 2a gegeben.Jeder Term der quadratischen Gleichung stellt eine physikalische Größe dar, die die Höhe des Balls beeinflusst.Der t-Quadrat-Term spiegelt die konstante Beschleunigung durch die Schwerkraft wider, die den Weg nach unten krümmt.Der Koeffizient des linearen t stellt die Anfangsgeschwindigkeit dar, die bestimmt, wie schnell die Kugel steigt.Der konstante Term stellt die Anfangshöhe dar, aus der der Ball geworfen wird.Unter Verwendung der aktualisierten Koeffizienten ergibt das Einsetzen des Zeitwerts in die Funktion den höchsten Punkt des Fluges des Balls.

quadratische modelle

Quadratische Modelle sind mathematische Darstellungen zur Beschreibung von Beziehungen, deren &#196;nderungsrate sich mit konstanter Rate &#228;ndert. Diese Modelle treten in einer Vielzahl nat&#252;rlicher und technischer Systeme auf, insbesondere in solchen, die Bewegung, Kr&#228;fte und Optimierungsprobleme betreffen. Eine h&#228;ufige Anwendung ist die Analyse der vertikalen Bewegung von Objekten unter dem Einfluss der Schwerkraft, beispielsweise eines in die Luft geworfenen Balls.
In solchen Szenarien ver&#228;ndert sich die H&#246;he des Objekts im Zeitverlauf in einem gekr&#252;mmten Verlauf, steigt bis zu einem Maximum an und f&#228;llt anschlie&#223;end wieder ab. Dieser gekr&#252;mmte Verlauf spiegelt den konstanten Einfluss der Schwerkraft wider, die den Aufstieg des Objekts verlangsamt und seinen Abstieg beschleunigt. Die Form dieser Bewegung wird durch eine Parabel beschrieben, ein typisches Kennzeichen des quadratischen Verhaltens.
Der H&#246;hepunkt der Bewegung &#8211; wenn das Objekt seinen h&#246;chsten Punkt erreicht &#8211; entspricht dem Scheitelpunkt der Parabel. Dieser Punkt markiert den &#220;bergang von der Aufw&#228;rts- zur Abw&#228;rtsbewegung und ist f&#252;r das Verst&#228;ndnis der Gesamtbahn unerl&#228;sslich.
Quadratische Modelle sind nicht nur f&#252;r die Interpretation von Bewegungen wertvoll, sondern auch f&#252;r die L&#246;sung von Problemen in Bereichen wie Physik, Ingenieurwesen, Wirtschaft und Biologie. Sie bieten Einblicke in Systeme, in denen Ver&#228;nderungen in einem vorhersehbaren, gekr&#252;mmten Muster auftreten, und erm&#246;glichen pr&#228;zise Analysen und Prognosen.

Quadratische Modelle

Quadratische Modelle sind mathematische Darstellungen zur Beschreibung von Beziehungen, deren &#196;nderungsrate sich mit konstanter Rate &#228;ndert. ...

Education

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JoVE Core

Math Fundamentals

Mathematische Grundlagen

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Polynomial and Rational Functions

Polynomfunktionen und rationale Funktionen

KEY TERMS AND CONCEPTS

KERNBEGRIFFE UND KONZEPTE

Quadratic Models

quadratic motion and parabolas

Quadratic models describe relationships where the rate of change changes at a constant rate. In math and science, this often shows up as a curved pattern called a parabola. These models help explain how a system changes over time or under a set force.

A common example is the vertical motion of an object moving under gravity, such as a ball thrown into the air. The object rises to a maximum height and then falls back down. Gravity slows the ascent and speeds up the descent, which creates the curved path seen in a quadratic model.

The highest point of that path is the turning point of the parabola. At this point, the motion changes from upward to downward. Knowing the turning point helps students describe the full trajectory of the object.

Quadratic models are also useful in physics, engineering, economics, and biology. They support analysis in systems where change follows a predictable curved pattern. That makes them useful for understanding results and making forecasts in many real-world situations.

Quadratic Motion and Parabolas

quadratic models

Quadratic models describe relationships where the rate of change changes at a constant rate. In math and science, this often shows up as a curved pattern called a parabola. These models help explain how a system changes over time or under a set force. A common example is the vertical motion of an object moving under gravity, such as a ball thrown into the air. The object rises to a maximum height and then falls back down. Gravity slows the ascent and speeds up the descent, which creates the...

introduction to polynomial functions

Polynomial Graphs, Zeros, and Degree

Polynomial functions are an important part of algebra and calculus. They are made from variables and constants combined by addition, subtraction, and multiplication. The variable is raised to a nonnegative integer exponent. A general polynomial function of degree n has a leading term aₙxⁿ, where aₙ is not equal to 0. The coefficient aₙ is the leading coefficient, and a₀ is the constant term. Polynomials are grouped by degree. A constant function has degree zero, a linear function has degree...

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Using Synthetic Division to Find Factors

Synthetic division is a fast way to divide a polynomial by a linear binomial of the form x - c, where c is a real number. It gives a shorter method than polynomial long division and is useful for checking values and finding factors. The setup starts with the coefficients of the polynomial written in descending order of powers. If a term is missing, a zero must be used to keep the columns lined up. The constant c from the divisor x - c is placed to the left of the synthetic division table. The...

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Polynomials can be analyzed by finding their real zeros, which are the values of x that make the polynomial equal to zero. These points are also the x-intercepts of the graph. Understanding them helps students see where a polynomial crosses the x-axis. The Rational Zeros Theorem gives a list of possible rational solutions for a polynomial with integer coefficients. If f(x)=anx^n+...+a0, then every rational zero has the form p/q. In that form, p divides the constant term a0 and q divides the...

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Polynomial Roots in Complex Numbers

Polynomial roots in complex numbers are a key result in algebra. The Fundamental Theorem of Algebra says that every non-constant polynomial with complex coefficients has at least one complex zero. For a polynomial of degree n, where n is at least 1 and the leading coefficient a n is not 0, there is at least one solution in the complex number system. This result also applies to polynomials with real coefficients, because the real numbers are a subset of the complex numbers. The Complete...

complex zeros

Finding Polynomial Roots With Complex Numbers

Complex numbers help describe every root of a polynomial equation. A complex number has the form a + bi, where a and b are real numbers and i is the imaginary unit, with i^2 = -1. A complex zero is any value that makes P(x) = 0 for a polynomial P(x). The complex number system includes the real numbers, so it gives a complete way to study polynomial roots. Every polynomial of degree n  1 can be written as a product of n linear factors over the complex numbers. In that factorization, each c_i...

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Graphing Rational Functions with Asymptotes

Rational functions are quotients of two polynomials, with the denominator Q(x) not equal to zero. Their graphs often include asymptotes, which are lines the graph gets close to but does not touch. These lines help show how the function behaves near certain input values and at large values of x. A vertical asymptote appears when the denominator is zero and the numerator is not zero. At those x-values, the function is undefined. To find a vertical asymptote, solve Q(x) = 0. For example, a...

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Linear and nonlinear inequalities help describe variable relationships and the ranges that satisfy a condition. A linear inequality uses variables raised only to the first power. Its graph is a straight line that divides the coordinate plane into two regions. One region satisfies the inequality, and the other does not. Nonlinear inequalities use variables raised to powers greater than one. These include quadratic inequalities and higher-degree polynomial inequalities. Their graphs are curved,...

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Solving Nonlinear Inequalities on a Number Line

Nonlinear inequalities can be solved by checking where an expression is positive or negative on a number line. These inequalities often include squares, products, or variables in the denominator, so they do not follow the same simple pattern as linear inequalities. The first step is to rewrite the inequality so that zero is on one side. After rewriting, factor the expression if possible. Factoring helps you find the key values where the expression equals zero. In one example, the values 3 and...

key terms and concepts

Learn how quadratic models describe curved motion, including a ball thrown into the air, and how the turning point shows the highest point of a parabola.

Quadratic Motion and Parabolas | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

math fundamentals

Quadratic Motion and Parabolas

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