Name: finding car stopping distance with integration
Uploaded: 2026-01-12T12:24:22.000Z
Duration: 1 min 26 s
Channel: JoVE

A derivative describes how a quantity changes with respect to its input variable. An antiderivative reverses this, recovering the original function from its rate of change.This reverse process is called integration. It helps find velocity from acceleration and displacement from velocity.Imagine a car moving steadily at 20 meters per second, and an obstacle appears 800 meters ahead of it.To stop before reaching the obstacle, assume the car slows down with a constant acceleration a. Integrating this acceleration gives the velocity as a function of time.The integration constant is found to be equal to the initial velocity.Next, integrating the velocity gives the displacement function. As the initial position is zero, the integration constant is also zero.Since the final velocity of the car is zero, substituting this value into the velocity equation gives the stopping time.Using this time in the displacement function and setting the displacement equal to 800 meters gives an equation for the required acceleration.This shows how integration links acceleration, velocity, and displacement.

toepassing van primitieve functies: rechtlijnige beweging

Application of Antiderivatives: Linear Motion

Een afgeleide beschrijft hoe een grootheid verandert ten opzichte van een andere, bijvoorbeeld hoe de snelheid met de tijd verandert. Het omgekeerde proces, waarbij een grootheid wordt teruggevonden uit de veranderingssnelheid ervan, staat bekend als integratie. In de natuurkunde is integratie fundamenteel, omdat zij onderling samenhangende fysische grootheden met elkaar verbindt en versnelling, snelheid en verplaatsing als samenhangende aspecten van beweging beschouwt.
Beschouw een auto die met een constante snelheid van 20 m/s rijdt, terwijl er 800 m verderop een obstakel opduikt. Om tot stilstand te komen voordat het obstakel wordt bereikt, vertraagt de auto met een constante versnelling. Versnelling geeft aan hoe de snelheid van de auto in de tijd verandert. Door integratie toe te passen op deze constante versnelling kan de snelheid van de auto worden beschreven als een functie van de tijd. Deze functiebeschrijving bevat een constante die weerspiegelt dat de auto op het moment dat het remmen begint al een niet-nulsnelheid heeft, overeenkomend met de beginsnelheid.
Snelheid beschrijft hoe de positie met de tijd verandert. Door dezelfde redenering opnieuw toe te passen, kan de verplaatsing van de auto gedurende het reminterval worden beschreven. Ook deze stap introduceert een constante, die de beginpositie van de auto weergeeft. Indien de positie op het moment dat het remmen begint als nul wordt gedefinieerd, verdwijnt deze constante.
De auto komt tot stilstand wanneer de snelheid nul wordt. Deze voorwaarde bepaalt de duur van het remproces. De aldus verkregen remtijd kan vervolgens worden gebruikt om te berekenen hoe ver de auto aflegt tijdens het vertragen. Door te eisen dat deze afstand precies 800 m bedraagt, kan de grootte van de vereiste versnelling worden bepaald.
Dit voorbeeld laat zien hoe integratie een helder conceptueel verband legt tussen versnelling, snelheid en verplaatsing, en hoe veranderingen in beweging zich in de loop van de tijd opstapelen tot waarneembare resultaten.

Toepassing van primitieve functies: rechtlijnige beweging

Een afgeleide beschrijft hoe een grootheid verandert ten opzichte van een andere, bijvoorbeeld hoe de snelheid met de tijd verandert. Het omgekeerde ...

application-of-antiderivatives-linear-motion

Education

JoVE Core

Calculus

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Eenheid

Applications of Differentiation

Toepassingen van differentiatie

KEY TERMS AND CONCEPTS

BELANGRIJKE TERMEN EN CONCEPTEN

finding car stopping distance with integration

Integration connects acceleration, velocity, and displacement in linear motion. It works as the reverse of a derivative, which tells how one quantity changes with another. In physics, this makes it possible to move from a rate of change back to the motion itself.

A car provides a clear example. It is moving at a steady 20 meters per second when an obstacle appears 800 meters ahead. To stop in time, the car slows with a constant acceleration. That acceleration tells how the car’s velocity changes over time.

Using integration, the constant acceleration can be used to write velocity as a function of time. This expression includes a constant because the car already has an initial velocity when braking begins. Velocity then gives the rate of change of position. A second integration describes the car’s displacement during the braking interval.

This second step also includes a constant for the starting position. If position is set to zero at the moment braking starts, that constant disappears. The car stops when its velocity reaches zero. That stopping time can then be used to find how far the car travels while slowing down.

Finally, the braking distance is set equal to 800 meters. That condition allows the needed acceleration to be determined. The example shows how integration links acceleration, velocity, and displacement in a real motion problem.

Finding Car Stopping Distance with Integration

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application of antiderivatives: linear motion

Integration connects acceleration, velocity, and displacement in linear motion. It works as the reverse of a derivative, which tells how one quantity changes with another. In physics, this makes it possible to move from a rate of change back to the motion itself. A car provides a clear example. It is moving at a steady 20 meters per second when an obstacle appears 800 meters ahead. To stop in time, the car slows with a constant acceleration. That acceleration tells how the car’s velocity...

key terms and concepts

Use integration to connect acceleration, velocity, and displacement in a car braking problem and find the acceleration needed to stop 800 meters ahead.

Finding Car Stopping Distance with Integration | 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

Finding Car Stopping Distance with Integration

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