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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 6 - Applications of IntegrationProblem 6.1.38b
Chapter 6, Problem 6.1.38b

Deceleration A car slows down with an acceleration of a(t) = −15 ft/s². Assume v(0)=60 ft/s,s(0)=0, and t is measured in seconds.


b. How far does the car travel in the time it takes to come to rest?

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1
Identify the given information: acceleration \(a(t) = -15\) ft/s² (constant), initial velocity \(v(0) = 60\) ft/s, and initial position \(s(0) = 0\) ft.
Find the time \(t\) when the car comes to rest by setting the velocity \(v(t) = 0\). Use the relationship between acceleration and velocity: \(v(t) = v(0) + \int_0^t a(\tau) \, d\tau\).
Since acceleration is constant, express velocity as \(v(t) = v(0) + a \cdot t = 60 - 15t\). Solve for \(t\) when \(v(t) = 0\).
Find the position function \(s(t)\) by integrating the velocity function: \(s(t) = s(0) + \int_0^t v(\tau) \, d\tau\). Substitute \(v(\tau) = 60 - 15\tau\) and \(s(0) = 0\).
Evaluate \(s(t)\) at the time found in step 3 to determine the total distance traveled until the car comes to rest.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Kinematic Equations and Integration

Kinematic equations relate acceleration, velocity, and displacement over time. Since acceleration is given as a function of time, integrating acceleration yields velocity, and integrating velocity yields displacement. This process helps find how far the car travels before stopping.
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Initial Conditions in Motion Problems

Initial conditions like initial velocity v(0) and initial position s(0) are essential to solve differential equations uniquely. They allow determination of constants of integration, ensuring the solution matches the physical scenario of the car starting at 60 ft/s and position zero.
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Initial Value Problems

Time to Come to Rest

The time when the car comes to rest is when velocity equals zero. Using the velocity function derived from acceleration, setting v(t) = 0 and solving for t gives the stopping time. This time is then used to find the total distance traveled.
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Related Practice
Textbook Question

Different axes of revolution Suppose R is the region bounded by y=f(x) and y=g(x) on the interval [a, b], where f(x)≥g(x).


b. How is this formula changed if x0>b?

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Textbook Question

Emptying a cylindrical tank A cylindrical water tank has height 8 m and radius 2m (see figure).

b. Is it true that it takes half as much work to pump the water out of the tank when it is half full as when it is full? Explain.

Textbook Question

Emptying a conical tank A water tank is shaped like an inverted cone with height 6 m and base radius 1.5 m (see figure).

b. Is it true that it takes half as much work to pump the water out of the tank when it is filled to half its depth as when it is full? Explain.

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Textbook Question

A right circular cylinder with height R and radius R has a volume of VC=πR^3 (height = radius).


b. Find the volume of the hemisphere that is inscribed in the cylinder with the same base as the cylinder. Express the volume in terms of VC.

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


b. When the velocity is positive on an interval, the displacement and the distance traveled on that interval are equal.

Textbook Question

Determine whether the following statements are true and give an explanation or counterexample. 


b. If f is not one-to-one on the interval [a, b], then the area of the surface generated when the graph of f on [a, b] is revolved about the x-axis is not defined.