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

Imaginez une voiture circulant sur une autoroute droite. Sa position dans le temps est modélisée par une fonction dont le graphe est une courbe lisse.Cette fonction suit la cylindrée de la voiture par rapport à un point d’origine fixe au fil du temps.Le taux instantané de changement de position de la voiture est mesuré par sa vitesse, indiquant sa vitesse et sa direction. Elle se trouve en prenant la première dérivée de la fonction de position.La voiture s’arrête momentanément lorsque sa vitesse est nulle. Pour la fonction de position donnée, cette condition de vitesse nulle est respectée à des moments précis.Une vitesse positive indique que la voiture avance, tandis qu’une vitesse négative signifie qu’elle recule vers sa position initiale.De même, l’accélération mesure comment la vitesse évolue. Elle se trouve en prenant la dérivée de la fonction de vitesse.La voiture accélère lorsque la vitesse et l’accélération ont le même signe, et elle ralentit quand ils ont des signes opposésCela montre comment le taux de variation prédit l’état exact du mouvement de la voiture à un moment donné, au-delà de la simple observation.

application des taux de variation

Le mouvement d&#x2019;une voiture sur une autoroute peut &#xEA;tre analys&#xE9; &#xE0; l&#x2019;aide des principes fondamentaux de l&#x2019;analyse et de la cin&#xE9;matique. Au cours de son d&#xE9;placement, la position de la voiture &#xE9;volue dans le temps et peut &#xEA;tre repr&#xE9;sent&#xE9;e math&#xE9;matiquement en tant que fonction du temps. L&#x2019;analyse du taux de variation de ces grandeurs permet de mesurer la vitesse et l&#x2019;acc&#xE9;l&#xE9;ration, qui constituent des &#xE9;l&#xE9;ments fondamentaux de l&#x2019;&#xE9;tude du mouvement.
La vitesse d&#xE9;crit la mani&#xE8;re dont la position varie au cours du temps. La vitesse moyenne sur un intervalle de temps donn&#xE9; se calcule en divisant la variation de la position par la variation correspondante du temps. Cette mesure fournit une indication globale de la rapidit&#xE9; du d&#xE9;placement de la voiture, mais ne rend pas compte des variations &#xE9;ventuelles &#xE0; l&#x2019;int&#xE9;rieur de l&#x2019;intervalle consid&#xE9;r&#xE9;. Il est donc n&#xE9;cessaire de r&#xE9;duire progressivement la dur&#xE9;e de l&#x2019;intervalle afin de d&#xE9;terminer la vitesse exacte &#xE0; un instant donn&#xE9;, en tendant vers une valeur instantan&#xE9;e. Cette vitesse instantan&#xE9;e correspond &#xE0; la d&#xE9;riv&#xE9;e de la fonction de position et repr&#xE9;sente la pente de la droite tangente sur un graphique position-temps en un point donn&#xE9;.
L&#x2019;acc&#xE9;l&#xE9;ration mesure la mani&#xE8;re dont la vitesse varie au cours du temps. &#xC0; l&#x2019;instar de la vitesse, elle peut &#xEA;tre &#xE9;valu&#xE9;e sous la forme d&#x2019;une valeur moyenne sur une p&#xE9;riode donn&#xE9;e ou comme une valeur instantan&#xE9;e. L&#x2019;acc&#xE9;l&#xE9;ration moyenne s&#x2019;obtient en divisant la variation de la vitesse par la dur&#xE9;e &#xE9;coul&#xE9;e. Lorsque l&#x2019;intervalle de temps tend vers z&#xE9;ro, l&#x2019;acc&#xE9;l&#xE9;ration &#xE0; un instant pr&#xE9;cis est d&#xE9;termin&#xE9;e en d&#xE9;rivant la vitesse par rapport au temps. Cette d&#xE9;riv&#xE9;e seconde de la position par rapport au temps met en &#xE9;vidence le taux de variation de la vitesse de la voiture.
Les graphiques repr&#xE9;sentant la position et la vitesse en fonction du temps fournissent des informations essentielles sur le mouvement. Sur un graphique position-temps, la pente d&#x2019;une droite s&#xE9;cante correspond &#xE0; la vitesse moyenne, tandis que la pente de la droite tangente en un point donn&#xE9; indique la vitesse instantan&#xE9;e. Sur un graphique vitesse-temps, la pente de la droite tangente repr&#xE9;sente l&#x2019;acc&#xE9;l&#xE9;ration, illustrant ainsi la mani&#xE8;re dont la vitesse &#xE9;volue au cours du temps. Ces m&#xE9;thodes graphiques jouent un r&#xF4;le central en physique et en ing&#xE9;nierie, car elles permettent d&#x2019;interpr&#xE9;ter et de pr&#xE9;voir des situations de mouvement r&#xE9;elles.

Application des taux de variation

Le mouvement d&#x2019;une voiture sur une autoroute peut &#xEA;tre analys&#xE9; &#xE0; l&#x2019;aide des principes fondamentaux de l&#x2019;analyse et de ...

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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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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...

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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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