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

Uma derivada descreve como uma quantidade muda em relação à sua variável de entrada. Uma antiderivada inverte isso, recuperando a função original de sua taxa de variação.Esse processo reverso é chamado de integração. Ele ajuda a encontrar a velocidade da aceleração e o deslocamento da velocidade.Imagine um carro se movendo de forma constante a 20 metros por segundo, e um obstáculo aparece a 800 metros à sua frente.Para parar antes de chegar ao obstáculo, suponha que o carro desacelere com uma aceleração constante a. Integrando essa aceleração, a velocidade é obtida em função do tempo.A constante de integração é encontrada igual à velocidade inicial.Em seguida, integrando a velocidade obtém a função de deslocamento. Como a posição inicial é zero, a constante de integração também é zero.Como a velocidade final do carro é zero, substituir esse valor na equação da velocidade obtém o tempo de parada.Usando esse tempo na função de deslocamento e ajustando o deslocamento a 800 metros, obtém-se uma equação para a aceleração necessária.Isso mostra como a integração liga aceleração, velocidade e deslocamento.

aplicação de antiderivadas: movimento linear

Application of Antiderivatives: Linear Motion

Uma derivada descreve como uma grandeza muda em rela&#xE7;&#xE3;o a outra, por exemplo, como a velocidade varia ao longo do tempo. O processo inverso, que recupera uma grandeza a partir de sua taxa de varia&#xE7;&#xE3;o, &#xE9; conhecido como integra&#xE7;&#xE3;o. Em f&#xED;sica, a integra&#xE7;&#xE3;o &#xE9; fundamental porque conecta grandezas f&#xED;sicas relacionadas, permitindo que acelera&#xE7;&#xE3;o, velocidade e deslocamento sejam compreendidos como aspectos interligados do movimento.
Considere um carro que se desloca a uma velocidade constante de 20 m/s quando surge um obst&#xE1;culo a 800 m &#xE0; frente. Para parar antes de atingir o obst&#xE1;culo, o carro desacelera com acelera&#xE7;&#xE3;o constante. A acelera&#xE7;&#xE3;o indica como a velocidade do carro muda ao longo do tempo. Aplicando a integra&#xE7;&#xE3;o a essa acelera&#xE7;&#xE3;o constante, a velocidade do carro pode ser descrita como uma fun&#xE7;&#xE3;o do tempo. Essa descri&#xE7;&#xE3;o inclui uma constante que reflete o fato de que o carro j&#xE1; possui uma velocidade diferente de zero no momento em que a frenagem come&#xE7;a, o que corresponde &#xE0; sua velocidade inicial.
A velocidade descreve como a posi&#xE7;&#xE3;o muda com o tempo. Aplicando o mesmo racioc&#xED;nio novamente, pode-se descrever o deslocamento do carro durante o intervalo de frenagem. Essa etapa tamb&#xE9;m envolve uma constante, que representa a posi&#xE7;&#xE3;o inicial do carro. Se a posi&#xE7;&#xE3;o for definida como zero no instante em que a frenagem come&#xE7;a, essa constante desaparece.
O carro para quando sua velocidade chega a zero. Essa condi&#xE7;&#xE3;o determina a dura&#xE7;&#xE3;o do processo de frenagem. Esse tempo de parada pode ent&#xE3;o ser usado para avaliar a dist&#xE2;ncia percorrida pelo carro durante a desacelera&#xE7;&#xE3;o. Ao exigir que essa dist&#xE2;ncia seja exatamente 800 m, pode-se determinar a magnitude da acelera&#xE7;&#xE3;o necess&#xE1;ria.
Este exemplo demonstra como a integra&#xE7;&#xE3;o fornece uma liga&#xE7;&#xE3;o conceitual clara entre acelera&#xE7;&#xE3;o, velocidade e deslocamento, mostrando como as mudan&#xE7;as no movimento se acumulam ao longo do tempo para produzir resultados observ&#xE1;veis.

Aplicação de antiderivadas: movimento linear

Uma derivada descreve como uma grandeza muda em rela&#xE7;&#xE3;o a outra, por exemplo, como a velocidade varia ao longo do tempo. O processo inverso, que ...

application-of-antiderivatives-linear-motion

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Calculus

Cálculo

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Applications of Differentiation

Aplicações da Diferenciação

KEY TERMS AND CONCEPTS

TERMOS E CONCEITOS-CHAVE

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

finding-car-stopping-distance-with-integration

absolute and local extreme values

Finding Max and Min on Function Graphs

Absolute and local extreme values show the highest and lowest points on a function graph. Absolute maximum and absolute minimum values describe the highest and lowest values across the entire domain. Local maximum and local minimum values describe the highest and lowest values within a smaller part of the graph. Periodic functions such as sine and cosine can have extreme values at many points. Their repeating pattern creates multiple local extrema. In these cases, the derivative becomes zero...

critical numbers and the closed interval method

critical-numbers-and-the-closed-interval-method

Finding Absolute Extrema on a Closed Interval

Absolute extrema on a closed interval are found by checking critical numbers and the endpoints. A function’s maximum and minimum values help describe its overall behavior across its domain. These values are called extrema, and they can be local or absolute. Local extrema are peaks and valleys in a limited region. Absolute extrema are the highest or lowest points on the entire interval. Critical numbers are the domain values where the derivative is zero or does not exist. According to Fermat’s...

rolle’s theorem

Rolle’s Theorem and Zero Slope Points

Rolle’s Theorem explains when a differentiable function must have a flat tangent inside an interval. It applies to a real-valued function on a closed interval, which means the endpoints are included. The function must also be continuous on the whole interval and differentiable on the open interval, where the interior points lie. The theorem uses one more key condition. The function must have the same value at both endpoints. When that happens, Rolle’s Theorem guarantees at least one point...

the mean value theorem

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first-derivatives-and-the-shape-of-a-graph

Using First Derivatives to Find Graph Changes

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second derivatives and the shape of a graph

second-derivatives-and-the-shape-of-a-graph

Using Second Derivatives to Read Curvature

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second derivative test: problem solving

Classifying Critical Points with the Second Derivative

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Using Derivatives to Read a Graph

Derivatives help students read the graph of a function by showing where the graph rises, falls, and bends. The first derivative tells the slope of the tangent line at each point. The second derivative shows how the curve changes shape over an interval. Points where the first derivative is zero or undefined are critical points. These points are important because they may mark a local maximum, a local minimum, or no extremum at all. A local minimum appears when the first derivative changes from...

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slant asymptotes

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optimization problems

Pipe Length Around a Corner

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application of differentiation to business

application-of-differentiation-to-business

Maximizing Bakery Revenue with Calculus

Calculus helps businesses make pricing decisions that can improve revenue. In this example, a bakery uses a cupcake model to find the best price and daily sales volume. The goal is to see how changes in price affect demand and total revenue. The demand function shows the link between price and how many cupcakes are sold each day. In this model, p(x) gives the price per cupcake when x cupcakes are sold. Lower prices lead to higher demand, so the function captures how sales change with price. <br...

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Newton’s Method for Finding Roots

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the antiderivative of a function

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Antiderivative Graph Shape and Slope

Antiderivative graphs show how a function accumulates over time. In calculus, an antiderivative tracks that buildup, much like the changing position of a rolling ball as its velocity changes. The graph of the antiderivative reflects the total effect of the original function's values. The sign of the original function helps determine the antiderivative's direction. When the function is positive, the antiderivative rises because the accumulation grows. When the function is negative, the...

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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