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A particle Is moving with the given data. Find the position of the particle a(t) = 2t + 3, s(0) v(o) -8s(t)...

Question

A particle Is moving with the given data. Find the position of the particle a(t) = 2t + 3, s(0) v(o) -8s(t)

A particle Is moving with the given data. Find the position of the particle a(t) = 2t + 3, s(0) v(o) -8 s(t)



Answers

A particle is moving with the given data. Find the position of the particle.

$ a(t) = 2t + 1 $, $ \quad s(0) = 3 $, $ \quad v(0) = -2 $

So we see that we have a particle moving um with the given data and we want to find the position of the particle. So the velocity is be of tea or we'll call it F. Of X. Is equal to she squared minus three routine. Or three X. Will change this tax as well. This is our graph and we want to find the anti derivative anti derivative S. F. T. So we take the anti driver. This that's going to give us X cube over three and then the three fruit X. That's the same thing as three times X. To the one half. So the way that we would do that would be it would be uh minus three X. To the three halves over times two thirds. So it's gonna be minus two ax to the three halves plus C. Evaluating the position S. Of four equals eight. We get the eight third to see so have plus eight thirds. And then we see that jeep crime of X. He's going to end up giving us the same exact graph of F. Fedex, which means we evaluated this properly. So the anti derivative is G. Of X equals X cubed minus three minus two X 23 have plus 8/3 final answer.

We know that velocity is the integral or anti derivative of acceleration. Therefore, let's check the integral when we got 1/3 t cute. Remember, the exponent goes up by one and then we divide by the new explain it when we're taking in the rolls. Now that we've got that, we know that position is the integral velocity. Now, remember, as a 00 and we end up with B our constant equal zero now remember us of one is 1 12 comes one to the fourth, minus 2/3 times one cubed plus three times one squared, plus eight times one, which is said able to 20 therefore are constant A is 211 over 12. Therefore, a final equation as 50 is 1 12 t to the fourth, minus 2/3 T cubed murders three T squared plus 211 over 12 t

Okay. Remember that acceleration is the derivative of lost piece of lost ease, the integral or anti derivative of acceleration. They're for V. A T. Thea anti derivative would be negative co signed t with three sign of tea, plus our constant A. You can also give it C or D. Any name really works and then position is the integral velocity. It's a night of signed team, as three co signed t plus a our first constant times TV Plus B, which is our second constant. Now we know as of 00 therefore us of zero, we can play again plus eight times zero plus b said equal to zero. This gives us b is equivalent to three and remember us of two pious 12 so passive to pie his negative ton sign to pie here minus three co sign to pa for us too. Hi plus three is equivalent to 12. Therefore, to pie eh equals 12 therefore is equivalent to sex over pi. Therefore, a final equation is us of cheese is negative. 10 cider T minus three co sign of tea Put six over pi T plus three

You've given the following problem F T is equal to 10 sanity plus Torrico Snooty s zero is equal to zero on s of two parties. You go to 12 on your ass if I in that city. So the start off what we can You know that the anti derivative f T is velocity soaks the interview Accelerations lost a so we can sell for that. So we get negative time co sign of tea plus three sine of t plus c next swing to find the introductory tive of bossy to find for safety will get us negative turn So unity when I was three Cose energy plus c x plus d The next thing we never asked zeros He was there on us of two. Possible to talk Also set the phone Chancellor's for set so she should sterile So here in 1910 Nitzana zero minus three times coastline is Oh, close. See, Time's over Crusty. What should leave us was zero. Does he call to number three plus t? Do you know what date is equal to three Next we can plug in to part in 12 soon Number 12 is equal to negative turn sign cheap. I my sweet coastline of two pi plus to part time. See? Plus three. Yes. So you know that. Sign it, you pies Just zero co signed up to par will get us in one. So we have 12 equal to scroll down a little bit together a little bit more real. So we know that 12 is equal to biggest story close to privacy plus degree. And so then we get this race cancels, We're left with two Parsi people. 12. So we get C s equals 12/2 pi, which leaves us with C is equal to six. We're playing. So what we have to do is just plug it back to your original question capacity He would you know, you have turn sign of tea. My story Coastline of tea Waas six of the rope I t plus three. And that is our answer


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