Name: velocity and acceleration in motion graphs
Uploaded: 2026-01-12T12:24:34.000Z
Duration: 1 min 18 s
Channel: JoVE

Consider a car moving along a straight highway. Its position over time is modeled by a function whose graph is a smooth curve.This function tracks the car's displacement relative to a fixed origin point over time.The instantaneous rate of change of position of the car is measured by its velocity, showing its speed and direction. It is found by taking the first derivative of the position function.The car is momentarily stopped when its velocity equals zero. For the given position function, this zero-velocity condition is met at specific moments.A positive velocity shows that the car is moving forward, while a negative velocity means the car is moving backward toward its initial position.Similarly, acceleration measures how the velocity changes. It is found by taking the derivative of the velocity function.The car speeds up when velocity and acceleration have the same sign, and it slows down when they have opposite signsThis shows how the rate of change predicts the exact state of motion of the car at any given time beyond simple observation.

toepassing van veranderingsnelheden

De beweging van een auto langs een snelweg kan worden onderzocht aan de hand van kernprincipes uit de differentiaalrekening en de kinematica. Terwijl de auto zich voortbeweegt, verandert zijn positie in de loop van de tijd en kan deze wiskundig worden gemodelleerd als een functie van de tijd. Door de veranderingssnelheid van deze veranderingen te analyseren, kunnen snelheid en versnelling worden bepaald, fundamentele grootheden binnen de bewegingsanalyse.
Snelheid beschrijft hoe de positie met de tijd verandert. De gemiddelde snelheid over een bepaald tijdsinterval wordt berekend door de verandering in positie te delen door de overeenkomstige verandering in tijd. Deze grootheid geeft een algemeen beeld van hoe snel de auto zich verplaatst, maar houdt geen rekening met variaties binnen het gekozen interval. Om de exacte snelheid op &#xE9;&#xE9;n specifiek moment te bepalen, moet het tijdsinterval steeds verder worden verkleind, waarbij men een ogenblikkelijke waarde benadert. Deze ogenblikkelijke snelheid komt overeen met de afgeleide van de positiefunctie en vertegenwoordigt de helling van de raaklijn aan de positie-tijdgrafiek in een bepaald punt.
Versnelling meet hoe de snelheid in de tijd verandert. Net als snelheid kan versnelling worden beschouwd als een gemiddelde over een bepaalde periode of als een ogenblikkelijke waarde. De gemiddelde versnelling wordt verkregen door de verandering in snelheid te delen door de benodigde tijd. Wanneer het tijdsinterval naar nul nadert, wordt de versnelling op een specifiek moment bepaald door de snelheid te differenti&#xEB;ren naar de tijd. Deze tweede afgeleide van de positie ten opzichte van de tijd geeft de veranderingssnelheid aan waarmee de snelheid van de auto verandert.
Grafieken van positie en snelheid als functie van de tijd bieden waardevolle inzichten in de beweging. In een positie-tijdgrafiek stelt de helling van een secantlijn de gemiddelde snelheid voor, terwijl de helling van de raaklijn in een bepaald punt de ogenblikkelijke snelheid weergeeft. In een snelheid-tijdgrafiek vertegenwoordigt de helling van de raaklijn de versnelling, waarmee wordt getoond hoe de snelheid in de loop van de tijd verandert. Deze grafische analysemethoden spelen een essenti&#xEB;le rol binnen de natuurkunde en de techniek, doordat zij het mogelijk maken om bewegingssituaties in de praktijk te interpreteren en te voorspellen.

Toepassing van veranderingsnelheden

De beweging van een auto langs een snelweg kan worden onderzocht aan de hand van kernprincipes uit de differentiaalrekening en de kinematica. Terwijl de ...

Education

JoVE Core

Calculus

Unit

Eenheid

Differentiation Rules

Differentieregels

KEY TERMS AND CONCEPTS

BELANGRIJKE TERMEN EN CONCEPTEN

Application of Rates of Change

velocity and acceleration in motion graphs

Velocity and acceleration describe how a car moves along a highway. The car’s position changes over time, so it can be written as a function of time. Rates of change help measure motion in a clear mathematical way.

Velocity tells how position changes with time. Average velocity is found by dividing the change in position by the change in time. It gives the overall speed of the car during a time interval, but it does not show changes inside that interval.

To find the velocity at one exact moment, the time interval must get smaller and smaller. This leads to instantaneous velocity. It is the derivative of the position function and matches the slope of the tangent line on a position-time graph.

Acceleration describes how velocity changes over time. Like velocity, it can be measured as an average value over a time period or as an instantaneous value at a specific moment. Average acceleration is the change in velocity divided by the time taken.

When the time interval approaches zero, acceleration at one moment is found by differentiating velocity with respect to time. This is the second derivative of position with respect to time. It shows the rate at which the car’s velocity changes.

Graphs of position and velocity over time help interpret motion. On a position-time graph, the slope of a secant line shows average velocity, and the slope of a tangent line shows instantaneous velocity. On a velocity-time graph, the slope of the tangent line shows acceleration. These graph tools are useful in physics and engineering for understanding and predicting real-world motion.

Velocity and Acceleration in Motion Graphs

velocity-and-acceleration-in-motion-graphs

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the quotient rule

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limits on trigonometric functions

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implicit differentiation

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e from Natural Log Derivatives

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application of rates of change

Velocity and acceleration describe how a car moves along a highway. The car’s position changes over time, so it can be written as a function of time. Rates of change help measure motion in a clear mathematical way. Velocity tells how position changes with time. Average velocity is found by dividing the change in position by the change in time. It gives the overall speed of the car during a time interval, but it does not show changes inside that interval. To find the velocity at one exact...

exponential growth

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related rates

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linearization and approximation

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application of linearization and approximation

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hyperbolic functions

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inverse hyperbolic functions and their derivatives

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key terms and concepts

Learn how position-time and velocity-time graphs show average and instantaneous velocity, plus acceleration, using derivatives and slopes in car motion.

Velocity and Acceleration in Motion Graphs | 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

Velocity and Acceleration in Motion Graphs

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