Explanation Given: The acceleration of an object: a ( t ) = − k v ( t ) . Initial velocity of the object = 1000ft/s. Final velocity of the object = 900ft/s. Distance travelled by the object = 1200ft. Formula used : The exponential decay of a function is of the form y ( t ) = y 0 e − k t , where the initial amount is y 0 and k is rate constant. Calculation: Given the acceleration of an object is a ( t ) = − k v ( t ) . From the Ex38, it is obtained that d v = − k d s , where v is the velocity of the object and s is the distance travelled by the object. Integrate the above statement, ∫ d v = ∫ − k d s (1) Initial velocity and the final velocity of the object is 1000ft/s and 900ft/s respectively. Hence, the lower limit for the left hand side of the integration of the equation (1) is 1000. The upper limit for the left hand side of the integration of the equation (1) is 900. Distance travelled by the object=1200ft. Hence, the lower limit for the right hand side of the integration of the equation (1) is 0. The upper limit for the right hand side of the integration of the equation (1) is 1200. Substitute the limits in the integral, ∫ 1000 900 d v = ∫ 0 1200 − k d s Integrate the above integral to obtain the value of k , [ v ] 1000 900 = − k [ s ] 0 1200 900 − 1000 = − k [ 1200 − 0 ] 1200 k = 100 k = 100 1200 On further simplification, the value of k becomes, k = 1 12 Let, the initial displacement be y 0 . The displacement after time t is obtained by using above stated formula. That is, y ( t ) = y 0 e − k t y ( t ) = y 0 e − t 12 ( ∵ k = 1 12 ) Assume the object travels 1000 ft in t 1 sec, that is y ( t 1 ) = 1000

Single Variable Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)

3rd Edition

ISBN: 9780134996103

Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz

Publisher: PEARSON

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Chapter 7.2, Problem 51E

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To approximate: The time required for the deceleration to occur.

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Single Variable Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)

Ch. 7.1 - What is the domain of ln |x|?Ch. 7.1 - Quick Check 2 Simplify e ln 2x, ln (e2x), e2 ln x,...Ch. 7.1 - Prob. 3QC Ch. 7.1 - Prob. 4QC Ch. 7.1 - Prob. 1E Ch. 7.1 - Prob. 2E Ch. 7.1 - Evaluate 4xdx.Ch. 7.1 - Prob. 4E Ch. 7.1 - Express 3x, x, and xsin x using the base e.Ch. 7.1 - Prob. 6E

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To approximate: The time required for the deceleration to occur.

Solution Summary: The author calculates the time required for the deceleration to occur, based on the exponential decay of a function.

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