(a) Explanation Given: The function of position of the object at time t is x = s ( t ) . The velocity function of the object at time t is v ( t ) . The acceleration function of the object at time t is a ( t ) . Apply work energy theorem: The net work done ( W ) is equal to the change in the kinetic energy ( 1 2 m v 2 2 − 1 2 m v 1 2 ) . It is expressed as follows: W = 1 2 m v 2 2 − 1 2 m v 1 2 Here, v 1 is the velocity at x 1 and v 2 is the velocity at x 2 Calculation: Show the expression to find the work done W by the object to move from a to b . W = lim n → ∞ ∑ i = 1 n f ( x i ∗ ) Δ x = ∫ a b f ( x ) d x (1) Consider the object moves from the location x 1 to x 2 . Substitute x 1 for a and x 2 for b in Equation (1). W = ∫ x 1 x 2 f ( x ) d x (2) Take x = s ( t ) (3) Differentiate Equation (3) with respect to time t as shown below: d x d t = d d t ( s ( t ) ) (4) The differentiation of space function with respect to time gives velocity function. Substitute v ( t ) for the left hand side of Equation (4). d x d t = v ( t ) d x = v ( t ) d t (5) Substitute v ( t ) d t for d t and s ( t ) for x in Equation (2). W = ∫ x 1 x 2 f ( s ( t ) ) v ( t ) d t (6) The position function x is a function of time. The position of the object is x 1 at time t 1 and the position of the object x 2 at time t 2 . Substitute t 1 for x 1 and t 2 for x 2 in Equation (6). W = ∫ t 1 t 2 f ( s ( t ) ) v ( t ) d t (7) Compare Equation (1) and Equation (7) (b) To determine To find: To find the work done to hurl a 12-lb bowling ball at 20 mi/h .

Single Variable Calculus: Early Transcendentals, Volume I

8th Edition

ISBN: 9781305270343

Author: James Stewart

Publisher: Cengage Learning

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Chapter 6.4, Problem 31E

(a)

To determine

To find:

To show that $W=∫x1x2f(x)dx=∫t1t2f(s(t))v(t)dt$ and prove Work Energy Theorem.

(b)

To determine

To find:

To find the work done to hurl a 12-lb bowling ball at $20 mi/h$ .

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Chapter 6.4, Problem 30E

Chapter 6.4, Problem 32E

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Chapter 6 Solutions

Single Variable Calculus: Early Transcendentals, Volume I

Ch. 6.1 - Find the area of the shaded region.Ch. 6.1 - Prob. 2E Ch. 6.1 - Prob. 3E Ch. 6.1 - Prob. 4E Ch. 6.1 - Prob. 5E Ch. 6.1 - Prob. 6E Ch. 6.1 - Sketch the region enclosed by the given curves....Ch. 6.1 - Sketch the region enclosed by the given curves....Ch. 6.1 - Sketch the region enclosed by the given curves....Ch. 6.1 - Sketch the region enclosed by the given curves....

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To show that W = ∫ x 1 x 2 f ( x ) d x = ∫ t 1 t 2 f ( s ( t ) ) v ( t ) d t and prove Work Energy Theorem.

Solution Summary: The author shows the expression to find the work done W by the object to move from a to b.

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To find:

To show that $W=∫x1x2f(x)dx=∫t1t2f(s(t))v(t)dt$ and prove Work Energy Theorem.

To find:

To find the work done to hurl a 12-lb bowling ball at $20 mi/h$ .

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Chapter 6 Solutions

To show that W = ∫ x 1 x 2 f ( x ) d x = ∫ t 1 t 2 f ( s ( t ) ) v ( t ) d t and prove Work Energy Theorem.

Solution Summary: The author shows the expression to find the work done W by the object to move from a to b.

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To find: To find the work done to hurl a 12-lb bowling ball at 20 mi/h<m:math><m:mrow><m:mn>20</m:mn><m:mtext> </m:mtext><m:mtext>mi/h</m:mtext></m:mrow></m:math>.

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Chapter 6 Solutions

To find:

To find the work done to hurl a 12-lb bowling ball at $20 mi/h$ .