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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

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Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem

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.)

To determine

a)

To prove:

W=x1x2fxdx=t1t2fstv(t)dt

Explanation

1) Concept:

The work is calculated by using the formula W=x1x2fxdx

2) Calculation:

 Since the work done is calculated by the formula

W=x1x2fxdx

Therefore, to prove the work-energy theorem given that the position of the function is st  and velocity is v(t). Therefore, substitute x=st so, by differentiation dx=vtdt

Hence, the work the work done is

W=t1t2f(s(t))v(t)dt

Hence, it is proved.

By using Newton’s second law of motion

force=mass×acceleration 

f=m·a

The integral becomes

W=t1t2fstvtdt

W=t1t2ma(t)v(t)dt

Put vt=u, therefore, the differentiation gives atdt=

To determine

b)

To find:

The work required to hurl a 12-lb bowling ball at 20 mi/h.

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