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Question 6 [6 marks](Pc 2.1,2.2,2.5]An object is moving along the curve X =t 6t2 + St , where X is the distance in m and tis the time Find the velocity att = 3 secb...

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Question 6 [6 marks](Pc 2.1,2.2,2.5]An object is moving along the curve X =t 6t2 + St , where X is the distance in m and tis the time Find the velocity att = 3 secb) Find the acceleration at t = 7 sec:Show each step of your work

Question 6 [6 marks] (Pc 2.1,2.2,2.5] An object is moving along the curve X =t 6t2 + St , where X is the distance in m and tis the time Find the velocity att = 3 sec b) Find the acceleration at t = 7 sec: Show each step of your work



Answers

An object moves in a straight line, and its position, $s,$ in metres after $t$ seconds is $s(t)=8-7 t+t^{2}$. a. Determine the velocity when $t=5$ b. Determine the acceleration when $t=5$.

We want to use the displacement function minus squared on the intervals. Zero, two, six, two First find bodies, displacement and philosophy on our time interval. So let's first go ahead into So we want to find our displacement on our time interval. So this is paying for this place is going to be our total distance on our high level that we truly do So P s of six minus essence here, so plugging and six into six minus t square, we'LL give zero and then plug it and zero into their also. So we get our displacement and stuff. Just be zero. Our average velocity is going to be so average. All right, so average a loss. Well, this is equal to our change in our displacements who tell tow us over our change of time. Well, Delta Earth his ruling us what we found last time, our total displacement. So that's going to be plug this in. So the zero over and our change time r interval of sixty cereals with the over six so our average displacement will end up being our average. Rossi ends up being zero on this in trouble here. Now we want to find the body speed and acceleration at a bar in points. So first we need to remember. So hello, Hostility. Is it too good? Derivative? Oh, displacement lift, respected time. So let's go ahead and write this. So that's prime tea for an extra. Then you go ahead and scoot off this over a little bit. That's fine. And then this is over. And then I'll put flying down again right here. So as prime is going to be all right, this has six t t minus team. He squared. And so we can do this by you. Some slash constant property or derivatives and then taking the directive of each of those will be powerful sense t to the one. I wouldn't need this help front. Subtract one off of that six times one t to the zero and t to the zero. Just be one so we don't need that. And then minus thanks just to move it out front. One officer, he times two t to the first power. So we just write in this So we end up with our velocity function and being six minus two t. So we want to find r pinpoint values or velocity. So be of zero is going to be, well, that'LL just six and then be of six is going to be So let's go ahead and plug that it will be six minus. Well, so we end up with negative six. So p m zero it sticks on DH B of six is mega substance. And the only thing we'LL need to remember is that celebration, Daisy to the derivative of philosophy. So on skin, we can use the Salman constant property to rewrite our velocity, which we have here, as did too, of six minus two David T. And so again, this is gonna be by something constant property and then taking the guerrilla groups each thieves will beat So the derivative of a constant zero And then once again, the derivative T would just be one. So we end up with minus two here or our acceleration that's a T. And then we want to find the value for these in points. So hey, of zero, Well, there's nowhere for me to plug in t so this will be negative too. And likewise or six. This will be negative too. So the acceleration will be the same for every value on our interval. Okay, Now, the last thing we want to do is find when, if ever during that terrible does the body change direction. Tom, just gonna do this on the next page. So sixteen honesty squared. So we have a busy to Earth's. Actually, we don't want us in this case, so chain, so change direction off. This implies our velocity changes. Sign. So Well, we want to do is look at our velocity. We're just going to be six minus two teeth. So the team is it too? Six minus two teeth. And we want to see if this ever changes. Sign on. So change is fine on zero to six. Well, this is a line. So the line is going to look like this, so could've started six. And then it's gonna go down like that. So this is what our velocity function looks like. So we know it will change sign, But it all depends on if it's in this interval. So what we could do? Let's go ahead and set. Our velocity equals zero so we can find out what this point right here is. So where were crossing expect since adding two over we'LL get six is equal to two team on Dividing Over by to get three is equal to so three is in our interval here. So our function will change directions once time sequel to three. So he is a three. The body changes directions and that's all we need to find sources.

For this problem. We're being asked given this equation for our position here for part A what is the total displacement? So to do this because we're giving our position function from time 0 to 6 weaken Plunkett or a final minus initial. So for us of six, we can see that this would be 36 minus 36. And this would be equal zero for on this is our final position. So and then we need to find an initial patient position. So for this we would see that this is 00 and that this would people zero. So this is also initials are finalised minus our initial. We're just because zero for average velocity, we need a first. Fine. What? How to copy velocity. And for this you take the derivative of our position. Right? So by doing that, we find the fee of tea equals negative two team plus six. And then we would add a final closer initial Invite that guy too. So our final would be made 12 close six. Should being negative six than our initial would be zero plastics, which would be six. So final plus and it should be negative six plus six over to Mrs Old soils with zero. As for a part B, this problem is asking us first. What is the speed I'm gonna say s of not to represent our speed? And this is essentially just stops the value of our philosophy. Like these points 10 points. So first looking at zero, we can plug in zero into her glossy function. So that zero, um, if you like this in, we find out the evil six after that, is this interesting? 60 years for second, let's be a zero. Um, same thing with at six at six. Um, negative 12 with six possibility of negative hopeful six would be of six oxidize without six communities per second as well. So this would be the speeds of endpoints both six years per second. And for the second part of B, we're asking for the acceleration of the endpoints, So acceleration would be again the derivative of velocity. You take that over this value a of two equal name two, and then therefore it like, um, acceleration would be negative. Two meters per second squared percents. Weird that both in points and then this problem asked less. Okay, So when does this change direction? So we would have to look at the policy because the changing in directions were basically implied that our velocity changes signs positive, negative, negative, positive. And how we do that is, actually, just take beauty was *** to t post six and 70 Then you can move one thing, two sides of the neck, good part right here to the side. And then we could win the to t you know, six and t equals three. So being changed at three seconds.

So for this problem were given this equation for position right here and for part A, They're asking us. Okay. So what is it? A spokesman? What are their change in position or changing us right there and then to find this basically refining. That's fine on my son s initial according to our time frame, which is six seconds. Is your seconds so fine. That's from just plug in. Um, the value at six for two Gs finals would be 36 my state, six b, 1st 0 and for us initial plug in zero, we get zero gun. So changing position is zero meters. No. For the second part of part of a, um it's asking us what is our average velocity. So are our patrol asik unrefined, um, by our change in position, over change in time. And since we found our change of position right here, zero weaken. Just to buy this by our time, this again would give us zero. So this would be zero meters per second. Now moving on to part B, question asked us. Okay. At our endpoints. What is the speed and call? This s not. And what is our acceleration so defined our speed we're gonna We're gonna find basically the opposite of value of the glossy and to find a velocity looking at our position function, we can take the driven of this to find the function for velocity, our equation for it. It becomes six minus toothy. And at zero, this value is Zeer six meters per second and six the Celje six mice clove acts of the oxidize was also examined beginning six meters per second. However, far to find our value for acceleration, we would need to take the derivative of loss to do that. So taking the druid of this, we get that exploration singer too. And this would be in a or two year squared for both of these endpoints, regardless. Because really have that about you along this, um, along this 0 to 6 times go. All right. Lastly, Parsi So Horsey asked us. Okay, Is there a place over here where there's a change in direction and, um, to find this we can actually just set r b A t equals zero. And basically, is this six minus two g So their every move something to the side. We find that three t goes three seconds, right. However, this must be checked because so far really know that match three her body goes to There's a zero volume three. Cyril Ray. So the and check, for example, let's pull an opportunity. Merciful one but one. We find that this body has paused over here. However we pull out, say 56 minus 10 was negative. Its neck on the side, right? Starting with what? Our values greater than three. Therefore, yes, there is a change of direction. And that would be that, um t equals three seconds.

Try to find our velocity and acceleration functions. The velocity is the derivative of the position. So that will be six t squared minus 14 T plus four. And our acceleration is the derivative of the velocity. So that will be 12 t minus 14 Now, the acceleration after one second, which I will write is one will be 12 times one is 12 minus 14. Well, give us negative too Meters per second squared. I've brought in a graph with all three. No, we're position. Function here, as you can see, is a cubic. Our velocity is the parabola. And finally the acceleration is linear. And there they are on one graph.


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