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# The kinetic energy $KE$ of an object of mass m moving with velocity $v$ is defined as $KE = \frac{1}{2} mv^2$. 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: $\frac{1}{2} mv^2_2 - \frac{1}{2} mv^2_1$ , 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 (5.5.6) to show that $$W = \int_{x_1}^{x_2} f(x) dx = \int_{t_1}^{t_2} f(s(t)) v(t) dt$$The use Newton's Second Law of Motion (force = mass $\times$ 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, measure in slugs.)

## (a) $\frac{1}{2} m v_{2}^{2}-\frac{1}{2} m v_{1}^{2}$(b) $161 . \overline{3} \mathrm{ft}- \mathrm{lb}$

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Applications of Integration

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So start off by recalling the Newton's second law that war's equals the mask comes the acceleration. So ab equals M e. And, uh, where implied to use a substitution rule or into girl. So after the axe extends for the distance that this president say it's a function of time, right? So say, actually was passivity, huh? And by the news, the second low we know data it enables toe m times a lt and a here is Thea absolution function that the times he right. So now do you want to see to alphabets some city DT That's over work. So work. And that's his ass of tea. So it just the reward ass by x by It was, But it's really an ex by acid. Let's see here every quack in this part. We have tea once with T two Answer Constance, which sport for M and use that interview. We have a chance t a f t temps be of tea based upon last the amis acceleration. And here it was too m times v of t one to be of tea too. You Do you cause here we gonna d'oh u equals be of tea So d'you Lee goes to 80. The team that's about that the nation is acceleration A Is Devi over the tea. We just need to start applying all the detonation of exploration and velocity here. Okay, so after here, we have enough integral, but integrating it we're gonna have when I have the work. So it's integration. The workmen equals to the battle of his in the will. And that's what half of em the of tea to swear. Minus one happens em b a t one square. So that's how you prove over for mother for this up Connecticut energy. Okay, there's a party. So you are a And for part B, we're simply was simply just plugging in all the values. So nice phone there. Double eggs to 1/2 on B two squared, minus one hat on B one square. Well, fuck that in. We're gonna have our work equals to 1/2 tons through eight tons. 88 over three squared, minus zero square and valuables to once succeed, one times one is 113311333 Jule, Or say 161 to us. One of the three

University of Illinois at Urbana-Champaign

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Applications of Integration

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