two-racers-in-adjacent-lanes-move-with-velocity-functions-v_1t-mathrmm-mathrms-and-v_2t-mathrmm-math

Question

Two racers in adjacent lanes move with velocity functions $v_{1}(t) \mathrm{m} / \mathrm{s}$ and $v_{2}(t) \mathrm{m} / \mathrm{s}$, respectively. Suppose
that the racers are even at time $t=60 \mathrm{s}$. Interpret the value of the integral
$$
\int_{0}^{60}\left[v_{2}(t)-v_{1}(t)\right] d t
$$
in this context.

Gregory Higby · Accepted Answer

uh, obvious. Basically, how would show my work to explain this problem? Um, so that&#x27;s what they give you. And you have to explain. The work is if you were to actually work out this math, Um, well, the anti derivative of velocity is position. I like doing X, but some people do P, I&#x27;ll do p for because it&#x27;s just variable position. Um, my position of the first person, and that&#x27;s from 0 to 60. So as you plug in P two of 60 minus p one of 60 um, minus p two at time, zero whips, supposed to write down zero. And I just wrote down that, um minus the position of the first racer at zero. Now, here&#x27;s the thing Is they told you that at times 60 there, even, Well, if they&#x27;re even, like 100 m or David, look at the units. Um, if there, even in the subtraction is equal to zero. So basically, what it boils down to, um is we&#x27;re just left with this thing. Um, no, I guess I&#x27;ll distribute that minus in P to zero plus pew one of zero. Um, what it boils down to is, you know, we could switch this around, cause addition is community is, um What that equation shows us is the lead that the razor would have at that time. So, you know, based on Alex written this number minus this number. So you could say the first racer, um, has a lead. Well, I guess we could still be zero. So I need to be careful how I commit to that answer. But it shows you the first racers lead at T equals zero at time zero. And that&#x27;s really what? That equation represents our expression.

Ernest Castorena · Answer

this time, they tell us the Katie drives the choir that speed of 50 years ago. 75 with 10 co sign of 18 MPH. And Michael there is a car at a speed. GMT is equal to 50 close to 30 MPH and there on time team minutes per minute. They&#x27;re going at the speed given by the function. So suppose they are at the same location that Time T is equal to zero. Talents compete the in a row we get from zero to X when we subtract those two functions from each other. Well, that was what city could be. Five minus five. It&#x27;s fine. Yeah. Last 10 co sign of T my last 18. So they want us to compute it the 18 and interpret the integral in terms of a race between Kelly and Michael. Well, that&#x27;s what is the difference. The difference is their speed. When you take the integral of their different, that&#x27;s the same thing as the difference of the integral. So your computing in their relative positions, assuming they started with the same position, which we can. So if you&#x27;re subtracting the differences in their positions whenever this is greater than zero. Let&#x27;s see, who do we subtract from Katie? Whenever it&#x27;s greater than zero? Katie is lead whenever it is less than zero. Then Michael wasn&#x27;t really, And the moments when it&#x27;s equal to their is when they have caught up with each other and someone&#x27;s about to overtake the other, let&#x27;s see, should we compute the actual in their own? I suppose we could do that, but we&#x27;ve already done a fairly good interpretation. But if we did the second part of the fundamental turmeric Oculus to get the derivative of this well, if this is their position, then the derivative we&#x27;ll just tell us there relative speeds so he&#x27;s faster than they have in. Okay, sounds good. Well, that&#x27;s integrated anyway. OK, so we&#x27;ll have five to the X plus 10 coastline of X. Let&#x27;s see. Actually, let&#x27;s leave without see for now, T and then minus t squared, we&#x27;re gonna evaluate that X and zero Blahniks. We&#x27;re going to get what I was writing. My next plus 10 co sign X minus X squared. I don&#x27;t just attract evaluated at zero. What? We actually have something this time We have coastline of zero which is just the one. So we have ah, attractant you got. So our function is goes and collects plus final X minus, X squared minus. So if you were to compute, the zeros of us will be able to see you again because that Katie is in the lead. If it&#x27;s greater than zero Michael&#x27;s 1,000,000 it was less than zero by the Druids to find out when. How did you computed to its to know equal to the Kremlin? Yin&#x27;s method since we have a trade combined with Apollonia. But, yes, you can attract more information from this.

Gregory Higby · Answer

So, the way I&#x27;m reading this problem, it&#x27;s basically just asking you to evaluate plugging in 10 in four F 50 because that&#x27;s what FFT is minus. Now, G B t is this equation of 10 minus sign of tea? Uh, D t So as you distribute this negative in here, both these tens will cancel. But then that becomes the integral from zero to pi of sine of t d t eso. As we do this through the anti derivative of sign is negative co sign from zero to pi. So then you&#x27;re basically do a negative cosign of pi minus negative. So plus cosine of zero. Well, as you look at that, maybe you need familiarity with the unit circle. That cosign of pie is negative. One negative, negative one plus cosine of zero is one. So you get to So when you do the anti derivative of the difference of velocities, what you&#x27;re finding is position. Um so e guess what I&#x27;m getting at is for the context. Um, the different. What we found is the difference of positions. Different is two miles here, but then they also ask you to find the anti derivative of same thing from 0 to 2 pi. So they change colors. So now everything&#x27;s the same. Except they changed it to 0 to 2 pi. So all of my work will be the same. You know, all of this math is the same, except this number is now different. So now what? We have just gonna put equals for the sake of time. I hope everything I&#x27;m saying is making sense. You&#x27;re up around now to pie, plus cosign. Um zero. Well, if you look at cosign of two pi, what we get is negative one plus one, which equals zero. So the difference of positions now difference of positions zero. Essentially, what I&#x27;m saying is that the two runners air now tied in the race. Um, I think I&#x27;ve answered every question, but another question might pop up then is what about you know what if we did pie to two pi? Well, then you would get a negative because the only way for to go from two miles of a difference 20 miles of a difference is if the one runner caught up. So that&#x27;s why one of them would be negative. But, um anyway, I hope I&#x27;ve answered each question. That first part is different positions is two miles and then the second part is zero miles.

Shamshad Waris · Answer

room dead. Every city on the double three human hostility would be no less building, you know, minus building the northwest on also is equal. That Indian genius, Best city and physical. Did you? So I still can&#x27;t. It is so s o d Even it if you want. It&#x27;s up. Do you know? Plus guilty minus. It&#x27;s so do you manage ditty divided by you have less ability. My has Yeo Bless. Did you no other cancel that be, Do you? No. But like in the middle off. Good. The villa is a video Bless Guilty from here. Didn&#x27;t get minus one day. Do a Europe less DLD It&#x27;ll hold us back. Let&#x27;s see. Minus minus one day to a into you, Manus lt Going Respect less. See divided Wait to really know Simply find this one The nail yet minus one way to it. Applied the formula A plus Making up all this pet hair. Yes, crab plus b Expect unless to give me. So we can I hear minus one. They do a being the formula pedido squared plus Delfay, the whole is crap. Bless two in tow. 30 into delicate. Let&#x27;s see now. Here, minus minus Bless you. So you get a night less one day Do a applied the Parma layer a menace Pick it up this way So we like their PS Well plus del dia sweat minus two PS where until 20 del deep minus C year divided by to cancel the sea Because I wanted a place on one CD minus the minuses. So we get gets a lot not simply by this one. Then they get here. Take the comma one day, two weeks from here. So one day do it from May. Then you get minus t a square man and LPs squared minus two. He is where? Building now. Here plus T s wet. Blessed Guilty square managed to He s square building now Cancel TGS. Well, so guilty square from here. Divided by who did it? No. Simplify this one. Then they will get air a deace way, Deputy Not a two minus pool A p s. Where will be divided by two Billy now consultant to guilty from the numerator and denominator Also who from the numerator and denominator finally will get a piece Well now will appeal Indian genius Nasuti off their given puncher so yes. So in 10? Yes, physical toe. It&#x27;s That&#x27;s off the value of this desktop PC minus one day do into a applying the powerful of the derivative they get here poopy less the video of a constant Thomas. Do not cancel who? Then we&#x27;ll get minus 80 now. Love that at physical due to you. So a political to be you get 80 s Where a political group you we can see here It&#x27;s SRP evaded physical. Who s so de in the 10 years that Oh okay. You can see also a very minus 80 a square minus it 80 You there is also minus a you. So this one is angle.

Averell Hause · Answer

so to solve the problem, we&#x27;re going to note that the velocity is simply the derivative of the position function. So we can then say that the velocity of particle one would be equal to the derivative of the position of a particle one with respect to time. So this would be the derivative with respect to time of 6.0 t square plus three point 00 t and then plus 2.0 this is equaling 12 0.0 T plus 3.0 We can say that the velocity of particle to would be equal to velocity the initial velocity of particle to plus, uh, the derivative of the acceleration of particle to with respect rather the integral of the acceleration of particle. To with respect to time, this is equaling 20 0.0 plus the integral, uh, negative 8.0 t d too. And so this is equaling 20.0 minus 4.0 T square. We know that V one is the velocity of particle ones equaling the velocity of particle to and then we&#x27;re simply solving for the time so we can say 12.0 T plus 3.0 equals 20.0 minus 4.0 t squared. Uh, we know that then clear solving for four 0.0 t squared. So we&#x27;re moving this over, and then it will be a plus 12 0.0 t, and then it will be minus 20. So minus seven. Teen 0.0, this is equaling zero. This is a quadratic function, so we can simply use our solve function in your tea I 84 85 or 89 or to Seoul for this, and we find that tea is equaling 1.5 seconds. So at 1.5 seconds, uh, the velocity of the two particles will be equal to one another. And to find that velocity, the sub one equals visa to weaken. Simply plug t into any one of these equations. I&#x27;m going to choose the first equation because it&#x27;s simply easier. And we don&#x27;t have to find the square root of rather, we don&#x27;t have to find 1.5 seconds squared. So I&#x27;m just gonna choose the linear equation because calculating it would be a bit easier. So 12.0 multiplied by 1.5 plus 3.0 This is giving us 15.6 meters per second. So this would be the velocity, uh, where the velocities of the two particles are equal to another and this occurs AT T equals 1.5 seconds. That is the end of the solution. Thank you for watching.

Vinnu M · Answer

So in the given question, the position functions off. Duca&#x27;s A and B are given as follows. So in the first sub question, we need to find the how far Car E is ahead of carb even t zero. So when do you zero assail the feudal? His 20 and be of evil is Tzeitel. So see my US SB if equal to 20. So Qari is 20 units off distance. I heard off Karbi Now for the seven second portion. We need to find the instance off time when the cards are next to each other. Now for that SE minus SB must be equal to zero. So that means 15 D squared plus 10 t plus 20 minus phi T square miners 40 t must be equal to zero. So this goof senti squared minus 30 plus 20 equal cereal. So the fact that I think we have t minus one multiplied by TV minus two equals zero. So do excuse me time as one are to there should be no comma one or two. So at a time one and r two the guards have the same position. No, For a third and final question, we need to find the instant when they have the same velocity foot, that s dash ill be Must be Quito A stash Be off. He must be equal. So s tested. This means that velocity off a must people toe velocity off Piot An incident off time. That is, if our dance off t so this gives 30 de plus 10. It must be equal to then d plus 40 who disc used me 22 calling 30 sort De is three by two now to see which car is further, we need to substitute three by two inboard their position functions so far card A A sale three by two is equal toe 15 miners three by two whole squared minus 10 multiplied by three by two plus strange be for excuse me 38.75 No, for God be SB off three by two This five multiplied by three by two squared This is squared plus 40 multiplied by three by two You just me 71 0.25 So clearly God be Yes. Ah, Head off God

Problem

The accompanying figure shows acceleration versus…

Problem 45 Hard Difficulty

Answer

$$
t=0
$$

Related Courses

Calculus Early Transcendentals

Related Topics

Discussion

Top Calculus 2 / BC Educators

Grace H.

Catherine R.

Kristen K.

Joseph L.

Calculus 2 / BC Courses

Integration Review - Intro

Definite vs Indefinite

Recommended Videos

Watch More Solved Questions in Chapter 6

Video Transcript

Howard Anton, Irl Bivens, Stephen Davis

Calculus Early Transcendentals

Grace H.

Catherine R.

Kristen K.

Joseph L.

Integration Review - Intro

Definite vs Indefinite

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

$$t=0$$

Calculus Early Transcendentals

Grace H.

Catherine R.

Kristen K.

Joseph L.

Integration Review - Intro

Definite vs Indefinite

Howard Anton, Irl Bivens, Stephen DavisCalculus Early Transcendentals

Grace H.

Catherine R.

Kristen K.

Joseph L.

$$
t=0
$$

Howard Anton, Irl Bivens, Stephen Davis

Calculus Early Transcendentals