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University of North Texas

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Problem 75 Hard Difficulty

What constant acceleration is required to increase the speed of a car from $ 30\;mi/h $ to $ 50\;mi/h $ in $ 5 $ seconds?

Answer

$$
\approx 5.87 \mathrm{ft} / \mathrm{s}^{2}
$$

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Video Transcript

well, so if we want to increase the speed of this car from 30 to 50 miles in five seconds after 30 MPH to 50 MPH in five seconds, the first thing I would do is get all the by units in the same measurement, because knows, here we have miles. And but then time here we're having in seconds. So just to kind of make things a little bit easier, I'm going to convert this two hours. So to do that, we would need to divide despite 3. 60. So it be five or not. 30. 60 3000, 600. So five divided by 3600. That would simplify down to 1/72. So I'm just gonna leave it like that for right now, So this is ours. So now what we can do is since acceleration is just constant well, we know that acceleration is supposed to be the derivative of our time. I mean, the derivative of our velocity with respect to time. And so this is just a constant. We ask ourselves. Well, the anti derivative of this is going to be some function that we take the derivative it we just get a constant and well, that's going to be a of T. And then we would just add some arbitrary constant to this. Uh huh. And so now if we were to come over here and plug in T is equal to zero, we could solve for what see is because then that's going to tell us our velocity should be 30. So we have 30 is equal to so eight times zero is just zero, and then it just c So C is 30. So we have our velocity being a T plus 30. So now we just need to figure out Well, what is, um is supposed to be And we can use that using the fact that we know at five seconds or one over 720 hours, the velocity should be 50. So let's go ahead and write that so t equal to one over 720 philosophy should be 50. So we plugged that in. So 50 is equal to s 05 times a plus 30. I'm sorry. Not five, um, one over 720. So now we can go ahead, subtract So it be 20 is able to one over 728. Multiply that over. Um, so 7, 20 times 20 14,400 is our acceleration. And now remember the unit with this. So our time was in miles. I'm sorry, are distance is in miles and our time is in ours. So this is going to be MPH squared. Um, so that is one way to think about it. But normally we think of accelerations in terms of like, feet per second squared. Least we're going to be doing it in these units. So why don't we go in and convert that to the here also, So we have 18,000 MPH squared, so we would need to cancel out miles. So for every one mile, there is 5280 ft, and then the hours will need to cancel that out twice. So for one hour, there is going to be 3600 seconds and then one hour, 3600 seconds. So these units cancel out with each other. The miles cancel also that we just multiply straight across and then divide. So we multiply the numerator, and we get some huge number that we divide that by 3600 twice, and that gives 88/15 feet per second squared, which is approximately 5.87 So this here is going to be. And actually, you could have also used this one up here. So either one of these, Um, the important thing is that you actually write your units down with this, because otherwise, all right, it's kind of vastly different numbers, but as long as you have your units than it should be fine.