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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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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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Calculating work for different springs Calculate the work required to stretch the following springs 0.4 m from their equilibrium positions. Assume Hooke’s law is obeyed.

b. A spring that requires 2 J of work to be stretched 0.1 m from its equilibrium position

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In the design of solid objects (both artificial and natural), the ratio of the surface area to the volume of the object is important. Animals typically generate heat at a rate proportional to their volume and lose heat at a rate proportional to their surface area. Therefore, animals with a low SAV ratio tend to retain heat, whereas animals with a high SAV ratio (such as children and hummingbirds) lose heat relatively quickly.


b. What is the SAV ratio of a ball with radius a? 

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Textbook Question

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b. Find the displacement over the given interval. 


v(t) = 3t²−6t on [0, 3]

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b. Find the population P(t) at any time t≥0.

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b. The distance traveled between t=0 and t=5

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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.

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