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.

application of antiderivatives: linear motion

Application of Antiderivatives: Linear Motion

A derivative describes how one quantity changes with respect to another, such as how velocity changes over time. The reverse process, which recovers a quantity from its rate of change, is known as integration. In physics, integration is fundamental because it links related physical quantities, allowing acceleration, velocity, and displacement to be understood as connected aspects of motion.Consider a car traveling at a steady speed of 20 meters per second when an obstacle appears 800 meters ahead. To stop before reaching the obstacle, the car slows down with a constant acceleration. Acceleration indicates how the car&#8217;s velocity changes over time. By applying integration to this constant acceleration, the velocity of the car can be described as a function of time. This description includes a constant that reflects the fact that the car already has a nonzero speed at the moment braking begins, which corresponds to its initial velocity.Velocity describes how position changes with time. Applying the same reasoning again allows the car&#8217;s displacement to be described over the braking interval. This step also involves a constant, which represents the initial position of the car. If the position is defined as zero at the instant braking starts, this constant vanishes.The car comes to rest when its velocity reaches zero. This condition determines how long the braking process lasts. That stopping time can then be used to evaluate how far the car travels while slowing down. By requiring this distance to be exactly 800 meters, the magnitude of the required acceleration can be determined.This example demonstrates how integration provides a clear conceptual link between acceleration, velocity, and displacement, showing how changes in motion accumulate over time to produce observable outcomes.

A derivative describes how one quantity changes with respect to another, such as how velocity changes over time. The reverse process, which recovers a ...

application-of-antiderivatives-linear-motion

Education

JoVE Core

Calculus

Unit

Applications of Differentiation

KEY TERMS AND CONCEPTS

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

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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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