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A ball has pogition function r(t) = (et cog t, 2t gin t,et) for 0 < t < ln2. Find r (t)(b) Find the speed function of the ball.Find the distance the ball trav...

Question

A ball has pogition function r(t) = (et cog t, 2t gin t,et) for 0 < t < ln2. Find r (t)(b) Find the speed function of the ball.Find the distance the ball travels.

A ball has pogition function r(t) = (et cog t, 2t gin t,et) for 0 < t < ln2. Find r (t) (b) Find the speed function of the ball. Find the distance the ball travels.



Answers

The velocity at time $t$ seconds of a ball thrown up into the air is $v(t)=-32 t+75$ feet per second. (a) Find the displacement of the ball during the time interval $0 \leq t \leq 3$ (b) Given that the initial position of the ball is $s(0)=6$ feet, use (a) to determine its position at time $t=3$

We've been given a problem where the velocity of a ball is being thrown up into the air um at negative 32 t plus 75 ft per second. So again t is in seconds and velocity. This equation here will give us the velocity of the ball in feet per second. We have two things that we have to do. The first thing that we have to do is find the displacement of the ball on the interval from 0-3 seconds. The second thing we're going to have to do is actually find the position of the new position of the ball. Given that information from a if we know that the initial position of the ball is at six feet on the field or wherever they're throwing the ball from. Mhm. Okay, so background information, we have to remember what displacement means, displacement with particles or objects is the distance from the starting position. So um to find that we're going to use the definite integral of velocity. So the definite integral of this velocity or feet per second will give us the amount of feet that the ball has moved from the time it was thrown until the three seconds comes about. Okay. All right, let's go ahead and get started on a since we've been asked to find the displacement on the interval from 0 to 3, we know that the integral will go from 0 to 3 of our velocity function negative 32 T plus 75 DT Now we'll take our anti derivative. Mhm. We're going to add one to that exponents and multiply by the reciprocal there And evaluate from 0 to 3. Let's clean that up a little bit And start plugging in. Our upper bound minus are lower bound. Okay, so we've got negative 16 times three squared and we're gonna subtract plugging in that lower bound for limit in this part will be easy over here because it will just be zero negative 16 times nine happens to be negative 1 44 75 times three is 2 25. And then of course minus zero over here. So our final answer turns out to be 81 ft. Now you integrated velocity. So absolutely our label is 81 ft. So from the time the ball was thrown three seconds pass, the ball has moved 81 ft up into the air. Alright, let's look at B for B. They're asking us to find the actual position of the ball. Well, at time zero, it's been given that the ball was at six ft into whatever the playing field is. Okay, imagine it's on this this line on the playing field and six ft in. Okay, well if the ball was six ft in I could add however much movement there was, or displacement, Which is what we did with our 0-3 of VFTTT. And that should give me the new position. So we found our in our displacement in a It was 81. And so now we know that the new position of the ball is 87 ft.

Okay. Were given us of Tuz to 14 t squared, and we know that he is the seconds after the ball begins rolling part A were asked to find the average velocity of the falling of the bottle over falling time. Originals effort. I we have five 5.1 using our formula for average velocity we get B is equal to s city virgins 14 t squared minus s evaluated that, eh? Over a T minus eight. Okay, let's just do our two endpoints, actually. So that's 14 5.1 squared minus 14. Five squared over 5.1. It's five. What does that give us? It's 14 times 5.1 Wait When it's 14 times five square. Modify 5.1. When its wife That gives me one for one point. Sport report I we get five and points one. So we em equal to well, 5.1 squared times 14 minus 14 names. Five Squared over 5.1 minus five. What does that give us? 14 times my 0.1 squared minus 14 times my squared. What if by 5.11 it's fine. Use me for Eddie or 1 48 points twitching for I. Yeah. Five. Calm the fuck went 001 I'm not going to write the formula out. It's the same thing is easy. Dis plug in this. This is only the changes to a 14 tons. Five point, though. The one to part two minus 14 5 to 2, provided by 5.1 like That's why. Give me one for a job for you. So what? And it might be just five points. Oh, no, no one. We'll be, uh, musical. Two points for 14 times. Appoint one. No, there was no one Squares plays 14 times five squared. What if I five points? Also one in its place. This gives me one for general court. 0014 All right, Port B. We're asked to use our answers from parts A to draw a conclusion about things. Continuous velocity of the ball at Time T is equal to five seconds. Well, based on what we know, since they all follow something that's around 1 40 you can say when we have to use it with five are instantaneous. Velocity is around 1 40 1 40 feet

This is all number 19. Okay. Problem in the 19 in which we have been given, uh, the distance table by ball s t equal toe 10 t squired. Okay, it's estimated and feet. So we have to find balls every velocity from Let's start with a Okay. From Stephen. Equal toe 32 D two equal to four off Stephen equal to three toe. T two equal to four way. We have to find the average velocity. Yeah. Yeah. Okay. So let us take a look at the average velocity. If we have, uh, toe find every velocity from t want toe t two and distance traveled in time t even is this one. And from 32 is as too so every last days as to minus s one by t two minus Stephen. So this is the average velocity. Likewise, he is every is velocity equal to 10 and two for the squared minus standing too. Three square by four minus three. That is 160 minus 90 by one. That's 70 ft per second. So this is the average velocity in case A In case B, we have t even equal to three t to equal to 3.5. So in this case, every is velocity will be equal toe. Everybody lost is 10 and to 3.5 whole Squire minus 10 into three. Square by 3.5 minus three. So this will become, uh, 3.5 old Squire. So 12.25 key, that is 122.5 122.5 minus 90 by one, not 1015 So this will become 65 feet per second. Right? Okay. Similarly, in next case, See, we have t even equal toe three t two equal toe. 3.1 So every is velocity will be equal. Toe 10 in 23.1 whole square minus 10 in 23 whole square, about 3.1 minus three. Pay three points. Little button. Hold squared into 10 minus 19. Divide by. Okay, it is 0.6010 point 601 divided by 0.1 That is 60.1. Yeah, 60.1 feet per second. Okay, but day we have even equal toe three toe t to equal to three point 001 so average velocity will be equal toe 10 and 23.1 Hold Squire. Minors turning to three whole square developed by 3.1 minus three. So this is 3.1 whole square in 2 10 minus 90. That is 0.6001 thereupon 06001 divide by 0.1 Okay, 60.1 60.1 ft per second. So these are the velocity is every velocities. And let's just, uh, see one thing that I'm just I was just plugging in the value of tea in this equation to get these things. Thank you.

In this problem, we've been given a velocity equation, we have T equals negative 32 T plus 75. That's going to give us the velocity of a ball that's being thrown up into the air after t seconds and that will be in feet per second. All right. We have three jobs to do. First find the displacement on the interval from 1 to 3. Then we're gonna try to decide whether or not the ball is Above or below its position at time one, and then we're going to find the displacement again of the ball on the time interval from 1-5. All right. Let's recall that for finding displacement, displacement means the distance from the starting position uh to the ending position. So it tells us how far away from the start the particle or the ball in this case would be All right. So how do we find displacement? We find displacement by taking the anti derivative of velocity and using whatever Interpol has been given for our bounce. All right, let's go ahead and get ourselves started. So for a If we want to find the displacement on the interval from 1 to 3, we're going to have to find the definite integral from 1 to 3 of our velocity equation, which is negative 32 T plus 75. Alright. Finding art anti derivative. We're going to add one to that exponents and multiply by the reciprocal. So there's are adding one to the exponents. Multiplying by the reciprocal of the exponents And 75 t. evaluating this from 1 to 3. All right. That gives us -16 T square plus 75 T. On our interval from 123. Let's go ahead and plug and chug start plugging in our upper bound and then minus plugging in our lower found and nine times negative 16 is negative. one for 75 types three is 2 25. And then it looks like we've got plus 16 -75. When we combine all of those together, We end up with 22 ft. So the ball is 22 ft above where it started. That's what displacement means. This is positive. Okay. That's going to help us with our answer for B. So B is asking is the position of the ball at T equals three higher than it was at T equals one. Yes. Why? Because the displacement is positive. So that means that it would be above mm hmm. Where we started if the displacement was negative, it's telling us that the particle. Is that many feet below where we started? Okay. Let's go ahead and calculate are integral once again. But this time from 1-5 for C. And I'm just gonna write down V F T D. T. Because we know what that is from A. And we also already know what are anti derivative is. But this time we're evaluating from 1 to 5 instead of 123 Alright, plugging and chugging minus that lower limit. And we end up with negative 425 times negative 16 Plus 3 75 minus 16 minus 75 plus 16 minus 75. And that ends up giving us negative 84 ft. Which is an interesting answer because that means it is 84 ft below where it was at time equals one.


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