#### To determine

**a)**

**To prove:**

W=∫x1x2fxdx=∫t1t2fstv(t)dt

#### Answer

**W=∫t1t2matvtdt and W=12mv22-12mv12**

#### 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=du and the limits of u are v(t1)=v1 to v(t2)=v2 Thus

W=∫v1v2m·u du

Integrating

=mu22v1v2

=mv222-v122

=12mv22-v12

W=12mv22-12mv12

**Conclusion:**

Hence it is proved W=∫t1t2ma(t)v(t)dt and W=12mv22-12mv12

#### To determine

**b)**

**To find:**

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

#### Answer

W=161.33 ft-lb

#### Explanation

**1) Given:**

The velocity of a ball is 20 mi/h. and the mass of the ball is 12-lb.

**2) Calculation:**

By the given instruction to measure the mass in slugs,

Divide the given quantity of mass by 32 ft/s^{2}

Therefore, mass m=1232=38 slugs.

Now convert the velocity to ft/s^{2}

=20·5280·13600

=105636

=883 ft/s2

Now apply part a) W=12mv22-12mv12 with v1=0 and v2=883 and mass m=38

W=12388832-123802

W=11·443-0

W=161.33 ft-lb

**Conclusion:**

The work required to hurl a 12-lb bowling ball at 20 mi/h is W=161.33 ft-lb