Textbook Question
Work done by a spring A spring on a horizontal surface can be stretched and held 0.5 m from its equilibrium position with a force of 50 N.
b. How much work is done in compressing the spring 0.5 m from its equilibrium position?
Ch. 6 - Applications of Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 6, Problem 6.1.10c
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.
c. The position at t=5
Verified step by step guidance1
Identify the velocity function v(t) from the graph. Notice that from t=0 to t=3, the velocity is constant at 3 units, and from t=3 to t=5, the velocity decreases linearly from 3 to 0.
Recall that the position function s(t) is the integral of the velocity function v(t) with respect to time t, and the initial position is given as s(0) = 0.
Calculate the position at t=5 by finding the area under the velocity curve from t=0 to t=5. This area represents the displacement.
Break the area under the curve into two parts: a rectangle from t=0 to t=3 with height 3, and a triangle from t=3 to t=5 where velocity decreases from 3 to 0.
Compute the area of the rectangle as base times height: (3 - 0) * 3, and the area of the triangle as (1/2) * base * height: (1/2) * (5 - 3) * 3. Add these two areas to find the total displacement, which equals s(5).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Velocity and Position Relationship
Velocity is the rate of change of position with respect to time. To find the position at a given time, you integrate the velocity function over the time interval, starting from the initial position. This means the position is the accumulated area under the velocity-time graph.
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Derivatives Applied To Velocity
Definite Integral as Area Under Curve
The definite integral of a velocity function from time a to b represents the net displacement during that interval. Graphically, this is the area under the velocity curve between t = a and t = b, considering areas above the time-axis as positive and below as negative.
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05:43Definition of the Definite Integral
Piecewise Linear Functions
When velocity is given as a piecewise linear function, the area under the curve can be found by calculating areas of simple geometric shapes like rectangles and triangles. Summing these areas over the intervals gives the total displacement, which helps determine the position.
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05:36Piecewise Functions
Related Practice
Textbook Question
Probe speed A data collection probe is dropped from a stationary balloon, and it falls with a velocity (in m/s) given by v(t) = 9.8t, neglecting air resistance. After 10 s, a chute deploys and the probe immediately slows to a constant speed of 10 m/s, which it maintains until it enters the ocean.
c. If the probe was released from an altitude of 3 km, when does it enter the ocean?
Textbook Question
Oscillating growth rates Some species have growth rates that oscillate with an (approximately) constant period P. Consider the growth rate function N'(t) = r+A sin 2πt/P, where A and r are constants with units of individuals/yr, and t is measured in years. A species becomes extinct if its population ever reaches 0 after t=0.
c. Suppose P=10, A=50, and r=5. If the initial population is N(0)=10, does the population ever become extinct? Explain.
Textbook Question
Region R is revolved about the line y=1 to form a solid of revolution.
c. Write an integral for the volume of the solid.
Textbook Question
Probe speed A data collection probe is dropped from a stationary balloon, and it falls with a velocity (in m/s) given by v(t) = 9.8t, neglecting air resistance. After 10 s, a chute deploys and the probe immediately slows to a constant speed of 10 m/s, which it maintains until it enters the ocean.
b. How far does the probe fall in the first 30 s after it is released?
Textbook Question
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) = 4t³ - 24t²+20t on [0, 5]