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Ch. 6 - Applications of Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh. 6 - Applications of Integration문제 6.1.37d
6장, 문제 6.1.37d

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


d. How long does it take the racer to travel 300 ft?

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1
Identify the given acceleration function: \(a(t) = 88\) ft/s², which is constant.
Since acceleration is the derivative of velocity, integrate \(a(t)\) with respect to \(t\) to find the velocity function: \(v(t) = \int a(t) \, dt = \int 88 \, dt\).
Use the initial condition \(v(0) = 0\) to solve for the constant of integration in the velocity function.
Next, integrate the velocity function \(v(t)\) with respect to \(t\) to find the position function \(s(t)\): \(s(t) = \int v(t) \, dt\).
Use the initial condition \(s(0) = 0\) to solve for the constant of integration in the position function, then set \(s(t) = 300\) ft and solve for \(t\) to find the time it takes to travel 300 ft.

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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
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3m

주요 개념

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Acceleration and Its Relationship to Velocity and Position

Acceleration is the rate of change of velocity with respect to time. Given acceleration a(t), velocity v(t) can be found by integrating a(t). Similarly, position s(t) is found by integrating velocity. Understanding these relationships allows us to move from acceleration to position over time.
추천 영상:
가이드 코스
06:15
Derivatives Applied To Acceleration

Initial Conditions in Integration

Initial conditions such as v(0) = 0 and s(0) = 0 provide specific values needed to solve the constants of integration when finding velocity and position functions. These conditions ensure the solution matches the physical scenario described.
추천 영상:
가이드 코스
05:03
Initial Value Problems

Solving for Time from Position Function

Once the position function s(t) is determined, solving for the time t when s(t) equals a given distance (300 ft) involves algebraic manipulation. This step finds the exact time required for the racer to travel the specified distance.
추천 영상:
가이드 코스
4:46
Adding & Subtracting Functions Example 1
관련 실천
교과서 질문

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


c. If the riders ride for 2 hr, who rides farther? Interpret your answer geometrically using the graphs of part (a). 

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교과서 질문

Use the region R that is bounded by the graphs of y=1+√x,x=4, and y=1 complete the exercises.


Region R is revolved about the y-axis to form a solid of revolution whose cross sections are washers.


d. Write an integral for the volume of the solid.

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교과서 질문

9–10. Velocity graphs The figures show velocity functions for motion along a line. Assume the motion begins with an initial position of s(0)=0. Determine the following.

d. A piecewise function for s(t)

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교과서 질문

9–10. Velocity graphs The figures show velocity functions for motion along a line. Assume the motion begins with an initial position of s(0)=0. Determine the following.

d. A piecewise function for s(t)

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교과서 질문

13–16. Displacement from velocity Consider an object moving along a line with the given velocity v. Assume time t is measured in seconds and velocities have units of m/s.


c. Find the distance traveled over the given interval.


v(t) = 3t²−6t on [0, 3]

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교과서 질문

Flying into a headwind The velocity (in mi/hr) of an airplane flying into a headwind is given by v(t) = 30(16−t²), for 0≤t≤3. Assume s(0)=0 and t is measured in hours.


c. How far has the airplane traveled at the instant its velocity reaches 400 mi/hr?

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