Name: estimating distance from velocity data
Uploaded: 2026-01-12T12:24:45.000Z
Duration: 1 min 26 s
Channel: JoVE

Stellen Sie sich einen LKW vor, der auf einer Autobahn fährt. Um die Entfernung zu bestimmen, die es während eines gegebenen Zeitintervalls t zurücklegt, sucht man die Fläche unter seinem Geschwindigkeits-Zeit-Graphen.Da die Geschwindigkeit des Lastwagens zu diskreten Zeitpunkten bekannt ist, kann dieser Bereich nicht präzise berechnet werden. Stattdessen wird die Fläche mit der trapezförmigen Regel angenähert. Dazu teilen Sie das gesamte Zeitintervall in n gleich große Teilintervalle, jeweils mit einer Breite von Δt.Bei dieser Methode sind die beiden Endpunkte jedes Teilintervalls durch eine gerade Linie verbunden, wodurch Trapezforme entstehen.Für ein gegebenes Teilintervall, das zwei vertikale Seiten hat, die die Geschwindigkeiten repräsentieren, und eine horizontale Basis für die Zeitbreite, ist die Fläche dieses Trapezs der Durchschnitt der Längen seiner beiden vertikalen Seiten multipliziert mit der horizontalen Entfernung.Addiert man die Flächen aller n Trapeze, erhält man eine Annäherung an die Verschiebung.Nach algebraischen Manipulationen und dem Lösen dieser Summe erscheinen die erste und letzte Geschwindigkeit einmal, während innere Geschwindigkeiten doppelt erscheinen, da jede zu zwei Trapezen gehört.Nach der Kombination ähnlicher Terme nähert sich der resultierende Ausdruck der Gesamtverschiebung über das angegebene Zeitintervall.

mittelpunkt- und trapezregel

Die Absch&#xE4;tzung der von einem Fahrzeug zur&#xFC;ckgelegten Strecke anhand seiner &#xFC;ber die Zeit gemessenen Geschwindigkeit stellt ein h&#xE4;ufiges Problem in der Physik und im Ingenieurwesen dar. Liegen die Geschwindigkeitsdaten nur in diskreten Zeitintervallen und nicht als kontinuierliche Funktion vor, kommen h&#xE4;ufig numerische Integrationsverfahren wie die Trapezregel zum Einsatz, um die Gesamtstrecke zu approximieren.
Die Trapezregel arbeitet, indem das gesamte Zeitintervall in mehrere gleich lange Abschnitte unterteilt wird. Innerhalb jedes Abschnitts werden die gemessenen Geschwindigkeiten an den Endpunkten durch eine Gerade verbunden, wodurch in einem Geschwindigkeits-Zeit-Diagramm ein Trapez entsteht. Jedes Trapez besitzt zwei parallele Seiten, die die Geschwindigkeit am Anfang und am Ende des Abschnitts repr&#xE4;sentieren, sowie einen Abstand dieser parallelen Seiten, der dem Zeitintervall entspricht.
Zur Ann&#xE4;herung der zur&#xFC;ckgelegten Strecke wird die Fl&#xE4;che jedes Trapezes berechnet. Diese Fl&#xE4;che entspricht dem Mittelwert der Geschwindigkeiten an den Endpunkten des Abschnitts, multipliziert mit der Dauer des Abschnitts. Die Summe der Fl&#xE4;chen aller Trapeze liefert eine Sch&#xE4;tzung der insgesamt &#xFC;ber den gesamten Zeitraum zur&#xFC;ckgelegten Verschiebung.
Bei diesem Verfahren werden die zu Beginn und am Ende des gesamten Zeitintervalls gemessenen Geschwindigkeiten nur einmal ber&#xFC;cksichtigt, w&#xE4;hrend die Geschwindigkeiten an den Zwischenpunkten zu zwei benachbarten Trapezen beitragen. Dieser systematische Ansatz f&#xFC;hrt zu einer zuverl&#xE4;ssigen Approximation der Verschiebung, insbesondere wenn die Anzahl der Intervalle ausreichend gro&#xDF; ist. Die Trapezregel stellt somit eine grundlegende Methode der numerischen Analysis dar und bietet eine praktikable M&#xF6;glichkeit, empirische Daten aus Bewegungs- oder anderen zeitabh&#xE4;ngigen Prozessen zu integrieren.

Mittelpunkt- und Trapezregel

Die Absch&#xE4;tzung der von einem Fahrzeug zur&#xFC;ckgelegten Strecke anhand seiner &#xFC;ber die Zeit gemessenen Geschwindigkeit stellt ein ...

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ZENTRALE BEGRIFFE UND KONZEPTE

Trapezoidal Rule

estimating distance from velocity data

Estimating distance from velocity data is a common use of the trapezoidal rule. In physics and engineering, velocity is often recorded at separate time points instead of as one continuous equation. Numerical integration gives a way to approximate the total displacement from those measurements.

The method divides the full time interval into equal parts. For each part, the velocities at the two endpoints are joined with a straight line on a velocity-time graph. That line creates a trapezoid. Each trapezoid has a vertical side for velocity and a horizontal base for the time interval.

To estimate displacement, the area of each trapezoid is found. This area depends on the average of the endpoint velocities and the length of the time segment. Adding the areas of all the trapezoids gives an estimate of the total distance traveled over the entire period.

The first and last velocity values are used only once. The values in the middle are shared by two neighboring trapezoids. This makes the method efficient for discrete data. It is especially useful when the number of intervals is large, and it provides a practical way to integrate measured motion data and other time-dependent processes.

Estimating Distance from Velocity Data

integration by parts: indefinite integrals

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Integration by Parts in AC Circuit Analysis

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Using Parts and Substitution for Area

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Integrals of secant and tangent powers are often solved by choosing a trig substitution that matches the integrand. The key idea is to use the parity of the exponents, which means whether a power is even or odd. A good substitution lets part of the integrand pair with the derivative of a trig function. When the power of secant is even, tangent is the better substitution variable. Since the derivative of tangent is secant squared, a factor of sec^2 x can be split off. The remaining even power...

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Riemann sums estimate the area under a curve when an exact definite integral is hard to find. This often happens when a function has no known antiderivative or cannot be written in a closed form. It also happens when the function comes from measured data instead of a formula. The method works by splitting the interval of integration into equal parts. Each subinterval gets a rectangle, and the rectangle height comes from the function value at one chosen point in that part. Adding the rectangle...

trapezoidal rule

Estimating distance from velocity data is a common use of the trapezoidal rule. In physics and engineering, velocity is often recorded at separate time points instead of as one continuous equation. Numerical integration gives a way to approximate the total displacement from those measurements. The method divides the full time interval into equal parts. For each part, the velocities at the two endpoints are joined with a straight line on a velocity-time graph. That line creates a trapezoid.

simpson's rule i

Estimating Area with Simpson’s Rule

Simpson’s Rule is a numerical integration method for estimating the value of a definite integral. It is useful when an exact antiderivative is hard or impossible to find. The method works by fitting a unique parabola through three equally spaced points on a curve and then integrating that quadratic over the interval. This approach can give a very accurate area estimate from only a few sampled values. It works well for many smoothly varying functions. That makes it a practical tool when a curve...

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improper integrals: discontinuous integrands

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Improper Integrals at an Endpoint

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key terms and concepts

Learn how the trapezoidal rule uses velocity-time data to estimate displacement by splitting time into intervals and adding trapezoid areas.

Estimating Distance from Velocity Data | 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.

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Estimating Distance from Velocity Data

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