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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 6 - Applications of IntegrationProblema 6.1.37e
Capitolo 6, Problema 6.1.37e

Acceleration A drag racer accelerates at a(t)=88 ft/s². Assume v(0)=0, s(0)=0, and t is measured in seconds.


e. How far has the racer traveled when it reaches a speed of 178 ft/s?

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1
Identify the given information: acceleration \(a(t) = 88\) ft/s² (constant), initial velocity \(v(0) = 0\) ft/s, and initial position \(s(0) = 0\) ft.
Since acceleration is the derivative of velocity, express velocity as the integral of acceleration: \(v(t) = \int a(t) \, dt = \int 88 \, dt\).
Integrate the acceleration to find velocity: \(v(t) = 88t + C\). Use the initial condition \(v(0) = 0\) to solve for \(C\), which gives \(C = 0\), so \(v(t) = 88t\).
Use the velocity function to find the time \(t\) when the velocity reaches 178 ft/s by solving \(178 = 88t\) for \(t\).
Next, find the position function \(s(t)\) by integrating the velocity: \(s(t) = \int v(t) \, dt = \int 88t \, dt\). Use the initial condition \(s(0) = 0\) to find the constant of integration, then evaluate \(s(t)\) at the time found in the previous step to determine the distance traveled.

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Relationship Between Acceleration, Velocity, and Displacement

Acceleration is the rate of change of velocity with respect to time, and velocity is the rate of change of displacement. Understanding how to integrate acceleration to find velocity, and then integrate velocity to find displacement, is essential for solving motion problems.
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Using The Acceleration Function Example 1

Integration of Constant Acceleration

When acceleration is constant, velocity can be found by integrating acceleration over time, resulting in a linear function of time. Similarly, displacement is found by integrating velocity, yielding a quadratic function. This process allows calculation of position and velocity at any time.
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Using The Acceleration Function

Using Initial Conditions to Determine Constants

Initial conditions like v(0)=0 and s(0)=0 provide values to solve for integration constants after integrating acceleration and velocity. Applying these conditions ensures the solution accurately models the physical scenario from the given starting point.
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Initial Value Problems Example 1
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Bike race Theo and Sasha start at the same place on a straight road, riding bikes with the following velocities (measured in mi/hr). Assume t is measured in hours.

Theo: vT(t)=10, for t≥0

Sasha: vS(t)=15t, for 0≤t≤1, and vS(t)=15, for t>1


f. Suppose Sasha gives Theo a head start of 0.2 hr and the riders ride for 20 mi. Who wins the race?

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e. Describe the position of the object relative to its initial position after 8 seconds.

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


d. A particular marginal cost function has the property that it is positive and decreasing. The cost of increasing production from A units to 2A units is greater than the cost of increasing production from 2A units to 3A units.

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Where do they meet? Kelly started at noon (t=0) riding a bike from Niwot to Berthoud, a distance of 20 km, with velocity v(t) = 15 / (t + 1)² (decreasing because of fatigue). Sandy started at noon (t=0) riding a bike in the opposite direction from Berthoud to Niwot with velocity u(t) = 20 / (t + 1)² (also decreasing because of fatigue). Assume distance is measured in kilometers and time is measured in hours.


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