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Given by the after t seconds straight line particle moving along each , part- covered [ measureMent feet) = the units of intenals [ [J,3.5]: (in must include time t...

Question

Given by the after t seconds straight line particle moving along each , part- covered [ measureMent feet) = the units of intenals [ [J,3.5]: (in must include time the distance _ You panticle , over Suppose +2(_ of the= fk) velocity function _ the average - Calculate particle whell of tlie veloeity (instantaneous) the Calculate

given by the after t seconds straight line particle moving along each , part- covered [ measureMent feet) = the units of intenals [ [J,3.5]: (in must include time the distance _ You panticle , over Suppose +2(_ of the= fk) velocity function _ the average - Calculate particle whell of tlie veloeity (instantaneous) the Calculate



Answers

A particle moving along a straight line has a velocity of $v(t)=t^{2} e^{-t}$ after $t$ sec. How far does it travel in the first 2 sec? (Assume the units are in feet and express the answer in exact form.)

Everyone. So what we have is the motion of a particle that is to find by this equation for position here such that the position is a function of time. Is it cubed? Minus 60 squared minus 15 T plus seven. What we're asked to solve for is the total distance traveled, the average velocity, the average speed and the instantaneous velocity at most 10 seconds. So what we're going to do first is go ahead and take our expression for s. Our position, which is to recall s, is equal to take minus 60 squared minus 15 T plus seven. And take the derivative of this. Just get our velocity. So we see that V is equal to De S DT, which gives us three D squared minus 12 T minus 15. So we can actually go ahead and solve for r instantaneous velocity here. So, by plugging in t equals 10 we'll see that the V Atiq with 10 is 1 65 ft per second. So that's one part of our answer. So we go ahead box that off next week and take the derivative of velocity to get acceleration and we can say that a is equal to DVD T, and this gives us that acceleration is a function of time. This 60 minus 12. Now we can plug in, uh, t equal to 10 and we'll see the acceleration to equal to 10 is equal to 48 feet per second squared. This is another part of our answer. We could get hood and box. That a swell. Now, given that this motion is governed by a polynomial to solve for our total distance traveled, we need to solve for when the particle change direction and also find the positive route so we can see by solving this equation for zero is that a positive route occurs when t is equal to five seconds. So therefore, what we can say that AT T equals zero s is equal to 7 ft. That he equals five s is equal to negative 93 ft and at our final time, point of T equals 10. This is equal to 257 ft. So by summing these up, we can find the total distance that has traveled. So we see that this goes from positive to negative back positive. So what this means is that we can solve for the total distance. They're saying 7 ft plus 93 ft times two because we're going down and back to 93 coming back to zero plus 257 ft. This gives us a total distance, traveled equal to 450 ft. So this will be another part of our final answer. Now, we're also asked us all for the average velocity, A T equals zero so called that average velocity is equal to Delta s all of adult T, and this will be equal to our displacement. So Delta Ass is 2 57 minus seven all over about the tea, which is 10. And this gives us average velocity 25 feet per se. And now we look at our average speed. And remember that average speed call it. The average is just total distance traveled all over Delta T. And this is equal to 450 all over 10 seconds, which gives us an average speed of 45 ft per second. And that is there a complete final answer

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Were given a velocity function over the interval 1 to 5 and were asked to find the displacement and the distance traveled. So displacement has just changed in location from where you are at the beginning, to where you are at the end. Where's the distance? Travel will be how far you actually traveled in case you went backwards or forward. Okay. So, to find the displacement, all you have to do is integrate or find the anti derivative from 1 to 5 of the velocity. Okay, so you get That's a parentheses there. T cubed over three minus. T squared over two minus 12 T from 1 to 5. I'm going to go ahead and get a common denominator. First six. So to take. Cubed minus three t squared minus 72 t so 2 50 minus 75 minus 3 60 minus two minus three minus 72. Minus 25. Minus 70 plus 73. So minus 1 12/6 or minus 56/3 can. What's the negative Mean? What means from where you started your back in the left hand direction. 56 3rd. Okay. Now, to find the distance traveled, we got to see if there are any places where the velocity changes. So we'll know whether the object turned around or not. So we're gonna take the velocity and we're going to factor it. I'm gonna set it equal to zero. So we find that the velocity equals zero AT T equals four, which is in the interval, 1 to 5. It's also zero negative three. But that doesn't make any sense. Or just going. Ignore that, um if you put in a number less than four into the velocity prop equation, you get a negative number, and if you put a number between four and five in there, then you get a positive number. So what it says is it's going to the left first, and then after four seconds or when time is four, it stops and turns around and goes back to the right so you can see that the displacement is not the same as the distance traveled. Okay, so the displacement is from here to here, and the distant traveled is here and here. So we gotta find out how far it goes in the first four seconds, and then how far it goes in the last second and then added together. So we need S of T, which is theano tie derivative cubed over three minus t squared over two minus 12 t plus some constant, which I'm gonna go ahead and get the common denominator so I can calculate faster. Just leave the plus C out there because it's going to subtract off. So we need s of one, which is 16 to minus three cups. That 72 there minus 72 plus c which we know is minus 73 6 plus c and then we need s a four. So where are you when time is 4. 1/6 1 28 minus 16 times 3 48 minus 2 88 plus E minus 208 60 plus c And then sf five, which is 1/6 um 2. 50 minus 75. Minus 3 60 plus scene. What? Which was 1 85 minus 1. 85. 6 plus c. Yeah. So when this object first starts out, it's at negative. 73 6. And then it goes back here to negative 2086 So let's see. 208 minus 73. That's 135. So from one second to four seconds, the distance traveled is 1 35 6 plus. Then in the next second, it goes from minus +286 and turns around and goes back to minus 1 85 6. So too late. Minus 1. 85. 33. No. 23 maybe would be better there. Alright, So altogether it travels 158 6 or 79 6. So the distance traveled is a positive number 79 6 and the, uh, displacement is negative. Negative. 56. 30.

We have developed selective banks and he off. He is a group he square minus Steve Manus to win. Go ahead of him. 12 50. They want to five. You know, you could find the displacement. This department progress in 15 The DDT, this is equal. He squared with a steel in the store. Uh huh. Thank you. Bye. Tree honesty Square by two Store one Good fight. Visible toe one by three My Cuban is want you restaurant? I have squared minus one square. Yes. So there the five minutes one is that tree 1 24. Mission to four. The story of board This is a country by three in stool minutes 48 for 1 24. Minus 36 minus one. Mhm. 44. The three guru. Uh huh. Well, Paris 20 on a strip. Basics by three. Next week. Took find distance. This is equal integration. 1 to 5. Absolutely off beauty beauty. No T square minus sti stubble. Remind us. Tool in the wilderness. Do you Minus or okay. Plus three. Uh huh. Absolute off VT. Well, they t square minutes. Stephen Stewart. He demons ruling. People's three discovered in the zero which in place de tous les Ethan minus three p is greater than four and physical minus off P square Manistee minuscule you p minus foreign people. ISS three is less than zero. Uh huh. Some place mhm minus three. Listen hysterical before so integration 1 to 4 The D Bt minus 20 minutes. Goto five the d Nay thi this is a cold integration wantedto no minus T square minus humanist tool duty this integration goto fight. He squared my esteem. Honest. Oh duty. This is a guru Manistee, you buy three honesty square by two Must Welte one toe this Did you buy three. Stay square by two Stoeltje who could fine physical of minus number 342 minutes One pupil and their management that in this good mor minus one this one day three by Cuban a spoon and this one going toe five square, minus four squared. We're still five minutes work. This is a quarter minus on the tree 63 and a strong the 24th I was born into a tree As when they treat on 25 Manage 64 minutes Something the 25 minutes 16 with one he could have minus 21 on this together 36. His mhm 61 by three minus nine by two. Let's do it. I know Hans. There is 79 by three Mhm.


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