 # The kinetic energy KE of an object of mass m moving with velocity v is defined as KE= 1 2 m v 2 If a force f ( x ) acts on the object, moving it along the x-axis from x 1 to x 2 , the Work-Energy Theorem states that the net work done is equal to the change in kinetic energy: 1 2 m v 2 2 − 1 2 m v 1 2 , where v 1 is the velocity at x 1 and v 2 is the velocity at x 2 . (a) Let x = s ( t ) be the position function of the object at time t and v ( t ) , a ( t ) the velocity and acceleration functions. Prove the Work-Energy Theorem by first using the Substitution Rule for Definite Integrals (4.5.5) to show that W = ∫ x 1 x 2 f ( x ) d x = ∫ t 1 t 2 f ( s ( t ) ) v ( t ) d t Then use Newton’s Second Law of Motion ( force = mass × acceleration ) and the substitution u = v ( t ) to evaluate the integral. (b) How much work (in ft-lb) is required to hurl a 12-lb bowling ball at 20 mi/h ? (Note: Divide the weight in pounds by 32 ft/s 2 , the acceleration due to gravity, to find the mass, measured in slugs.) ### Calculus (MindTap Course List)

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781285740621 ### Calculus (MindTap Course List)

8th Edition
James Stewart
Publisher: Cengage Learning
ISBN: 9781285740621

#### Solutions

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Chapter 5.4, Problem 31E
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