The displacement (in centimeters) of a particle moving back and forth along a straight line is given by the equation of motion s = 2 sin πt + 3 cos πt, where t is measured in seconds.
(a) Find the average velocity during each time period:
(i) [1, 2]
(ii) (1, 1.1]
(iii) [1, 1.01]
(iv) [1, 1.001]
(b) Estimate the instantaneous velocity of the particle when t = 1.
(a)
To find: The average velocity for given time periods.
Answer to Problem 8E
(i) The average velocity over the time interval [1, 2] is
(ii) The average velocity over the time interval [1, 1.1] is
(iii) The average velocity over the time interval [1, 1.01] is
(iv) The average velocity over the time interval [1, 1.001] is
Explanation of Solution
Given:
The equation of a motion
Formula used:
The average velocity over the time interval
Calculation:
Section-(i)
Obtain the average velocity over the time interval [1, 2].
Substitute
Substitute the respective trigonometric values and simplify the terms.
Thus, the average velocity over the time interval [1, 2] is
Section-(ii)
Obtain the average velocity over the time interval [1, 1.1].
Substitute
Substitute the respective trigonometric values and simplify the terms.
Thus, the average velocity over the time interval [1, 1.1] is
Section-(iii)
Obtain the average velocity over the time interval [1, 1.01].
Substitute
Substitute the respective trigonometric values and simplify the terms.
Thus, the average velocity over the time interval [1, 1.01] is
Section-(iv)
Obtain the average velocity over the time interval [1, 1.001].
Substitute
Substitute the respective trigonometric values and simplify the terms.
Thus, the average velocity over the time interval [1, 1.001] is
(b)
To estimate: The instantaneous velocity of the particle when
Answer to Problem 8E
The estimated instantaneous velocity when
Explanation of Solution
The average velocity over the time interval [1, 1+h] is,
The instantaneous velocity when
That is, when h approaches 0, then t is closer to 1.
Thus, the estimated instantaneous velocity when
Chapter 2 Solutions
Single Variable Calculus: Concepts and Contexts, Enhanced Edition
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