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A certain Chevrolet Corvette goes from 0 mi/h to $60.0 \mathrm{mi} / \mathrm{h}$ $(88.0 \mathrm{ft} / \mathrm{s})$ in $3.60 \mathrm{s}$. Assuming constant acceleration, how far (in $\mathrm{ft}$ ) does it travel in this time?

$158 \mathrm{ft}$

Calculus 2 / BC

Chapter 26

Applications of Integration

Section 1

Applications of the Indefinite Integral

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So we're told that this car goes from 0 to 88 ft per 2nd and 3.6 seconds. We're also told that it has constant acceleration. So if it goes from 0 to 88 ft per second in 3.6 seconds, that our initial velocity v 00 And so our equation for velocity was again V. Zero plus our acceleration times T. So we can plug in zero for V. Zero. So we just have our acceleration times T. And we're also told that this is a constant acceleration. So A is just gonna be a constant. So you can plug in the 00.3 point six comma 88 to find our acceleration. Since we're told that we're going 88 miles per sorry feet per second after 3.6 seconds. So you have 88 it's equal to a times 36. So then a is equal to 88, Divided by 3.6. and so if you plug this into a calculator, 88, divided by 3.6 is equal to 24 0.4 for repeating. And this is in feet per second squared. So now we know our velocity equation. V. is just going to be equal to 24 .44 repeating times, teeth. And so if we want to figure out how far this um car travels in the time that it was going from 0 to 88 ft per second, we're gonna have to find the displacement equation which is just the integral of our velocity equation. So we're going to take the integral of our velocity equation which is equal to the integral 24 44 repeating times. T times DT Just then just equal to 24-44 repeating Times, T. divided by two. And then we also have plus C. But since our initial displacement is going to be zero, when we plug in zero for um T. We're just gonna get sees equal to zero so we can let C. Equals zero in this case. So we have our displacement equation S. F. T. It's equal to 24.44 repeating times, T squared divided by two. And now we just need to plug in 3.6 40 to figure out how far our car our car actually traveled in feet. Um In the time I went from 0 to 88 ft/s. So let's plug in 3.6 24.44 repeating times 3.6 squared, divided by two. So if you plug this into a calculator 24.444 Repeating Times 3 6 squared. It's equal to about 316 8. And then if we divide by two We get this is equal to 158 0.4 ft. So it took this car 100 and 58.4 ft, to go from zero MPH to 60 MPH, or to go from zero ft per second to 88 ft per second.

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