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6. In the adjacent figure_ the variation of the speed V of a particle as function of time vlmls Specify the nature of motion b) Write the expression of the speed as...

Question

6. In the adjacent figure_ the variation of the speed V of a particle as function of time vlmls Specify the nature of motion b) Write the expression of the speed as a function of time_ Write the time equation(x)) of the motion of particle. (Take Xo 4m at t =Determine the position of the particle at t=1sec and att =4 sec_ a) Deduce the distance covered between =1 sec and [=4 sec f) Determine the average speed between t =1 sec and t =4 sec.tls)

6. In the adjacent figure_ the variation of the speed V of a particle as function of time vlmls Specify the nature of motion b) Write the expression of the speed as a function of time_ Write the time equation(x)) of the motion of particle. (Take Xo 4m at t = Determine the position of the particle at t=1sec and att =4 sec_ a) Deduce the distance covered between =1 sec and [=4 sec f) Determine the average speed between t =1 sec and t =4 sec. tls)



Answers

The position of a particle as a function of time is given by $x=(6 \mathrm{m} / \mathrm{s}) t+\left(-2 \mathrm{m} / \mathrm{s}^{2}\right) t^{2}$ (a) Plot $x$ versus $t$ for $t=0$ to $t=2 \mathrm{s} .$ (b) Find the average velocity of the particle from $f=0$ to $t=1 \mathrm{s}$. (c) Find the average speed from $t=0$ to $t=1 \mathrm{s}$.

So we're told that the distance of a particle from some fixed point can be given by the equation. S A T is equal to t square plus five t plus two. We're t is measured in seconds, and we want to find the average velocity of the particle over the time intervals going from 4 to 6 and 4 to 5. And then they want us to find the instantaneous velocity off the particle when time is able to four. So one of things will need to remember for this question is that velocity is equal to the change in our position over our change in time. So that's why we can use this average rate of change equation on S a T here. So we just need to plug everything in to our equations here. So for the average rate of change for the velocity from 4 to 6, this is going to be s of six minus s up, four over six, minus four. So it's got unplugged these in. So we're gonna have s of six. And this is equal to so 36 plus 30 plus two and adding all of that up that looks like you to give us 68. So So go ahead and replace this with 68 ends and s of war is going to be 16 plus 20 plus two, which looks like it gives us 38 so we can replace that with 38 there. So now if we were to subtract everything, So 68 minus 38 is 30 and then divided by two. That would be 15. And then what were the units were this to see if they tell us? Oh, so they don't actually say the units. We really can't put anything with it. It looks like So I was gonna say 15 and then over here from 4 to 5. So we're gonna follow the same thing that we did before. So it's gonna be s of five minus s for over five minus four. So we already know that s of four is 38. So we just need to figure out what is us of five. So us of five. So it's gonna be 25 plus 25 plus two. So that's 52. So then we would do 52 minus 38 which is 14 divided by one. So just be 14. That would be the average rate of change, or B. Now they want to find the instantaneous rate of change. At times you go four seconds. Well, we could just go ahead and follow this equation. We have appear, and I like to do this in steps. So the first thing you need identifies that are a is going to before because that's where we're interested in with the institutions rate of change. So I like toe first, go ahead and do s of so before, plus h. So it's going to plug four plus h into here So before plus H squared plus five times for plus H plus two. So expend this out. We should have 16 plus eight h most H squared plus 20 lost five h plus two. Now we can go ahead and at all the subs, we'd have 16 plus 20 plus to which is going to be 38 and then we're gonna have eight h plus five h that's gonna be plus 13 h and then lastly, we just have the H squared term right now. What we want to do for our next step is due s for plus H minus best of four. So it's for about what s a four is actually already found that over here we found s a four supposed to be 38 so we could just go ahead and plug that in directly, so there's gonna be 38 plus 13 h plus H squared minus 38 knows thes 30. It's just counts out with each other and were left with 13 h plus h squared. And then for the the last step, we're going to take the limit as h approach zero of s four plus H minus s of four all over h and kind of as a pro tip. Once you get to this point here, if, for whatever reason, whatever you plug into the numeric cannot be evenly divided by this agent the Dominator, then you may need to go back in check your ouch broke. Because at least when we're working with polynomial, they should always cancel out, and so knows here we're just left with 13 plus h, which we can apply this summer directly. Now that'll be 13 plus zero, which gives us 13. So we end up with that instantaneous, falsely being 13

So in problem 23 we have ah function to describes the movement of a particle. Such a cz X equals two 60 minors to t square and the first item we have to blot the graph off position versus time when the particle goes from Time T CO zero, open to Time T coast, two seconds. Okay, so we just need to find out the position of the particle when t close your and when t it goes to 20 equals zero and the position is easy. We are in the hurry in the orange of the movement and Wen t equals two. We have X equals two, six times to minors. Two times four. So acts is going to be 12 miners. Eight. This is a go too far meters. Okay, so here we have the position. The final position of the Article 20 equals two. And well, since we know that this is a second degree function because off the T Square, we know that this graph is going to be profitable and we just need to determine if the parable is decreasing or recent, and information about that we find in the constant there follows the T square. So since the constant is negative deplorable, it's going to be Dick recent. And this is the graph up onto T close. Two seconds, arbol we continue it, but the problem only own wants to find out into here. Okay, well, the second item. Second ison. We want to know what the average a lost city of the particle. When we go from time t co zeros to time me quot equals one again. Let's find out the position. Barter crew when t equals one. We have six times one miners, two times run. The position is six minus two just for okay, four meters. So we know that the average velocity is describe us that's putting on a collar. The average velocity hopes the average velocity is the difference in position. Divided by the difference your time. So we have the difference in position is just the final position that is far finer. Zero divided for the difference in time. Just one second. So the average velocity is for meters, first seconds and the average speed it's going to be do you third item. The average speed is just the difference and velocity divided by the difference in time. This is four miners, six divided by one. This is it was to minus two meters for second square. This is the final answer. Thank you for watching.

So problem 23. We have a function that describes the movement of a particle such as? Let's see X equals six T minus two T square. Okay. And the first item off the problem, we have to blot a graph for position versus time. When the particles goes from T crew zero, open to T equals two seconds. OK, so how we gonna do this First? The position when tea cozy. Here. 20 co zero article is the orange of the movements in a 2nd 1 the time equals two seconds. The position it's going to be six times two minus two times four. So position accede Coast too. 12 minus eight. Finally, the position's going to be for meters. Okay, so four meters. Okay, but we can put someone something around here. Time t goes to positions here. This sis, the final position of the particle. Okay, so how we gonna make the whole graph? We know that this function is a second degree function and us all second degree functions. The graph is a pardonable. And we know this because we know this is a second degree function because of the T square in here, and we know that all second will give the grief functions The constant that follows that C square gives us the concussive ity of the horrible. Since this constant is negative, the comm cover T is going to be precent. So this is the graph off this horrible for this movements. Okay, this is the first item. The second item. We want to know the average velocity and we know that average velocity is defined a difference in position, divided by different sync time. This time we have to calculate the average velocity. When the particles goes from Chico's here upon to t equals one second. And I can we need to find out where is the position off The particle 20 equals one. So acts is going to be six times one miners, two times one. So the position is just two. Oh, sorry. Far position is for okay for meters. Fine. Let's go. The average velocity is just a difference in position. So four miners zero divided by the difference in time. One second. So the average velocity is just for meters. First seconds and the final item. We have to calculate the average speed off the particle. The average speed before recalculate this. We just need to know that a general function that describes the movement of a particle accelerated is X of time. Is it good to X zero? That is the initial position, plus the initial velocity times time, plus the acceleration of the divided by two times D square. Okay. And just look for the function we have here we have for this problem. Exit T equals 26 two minus two T square. So we just need to compare the argument of the square, the argument of the square and the function we have. So we have here a divided by two because mine is too. So the acceleration off the average speed is just four minus four meters per second square. This is the final answer. Thank you for watching

Or go for this problem. Um And um for the following problem is to consider the position function given to us, which in this case is S. F. T equals t square. I have three peoples too. And with this model we can find so much information. We see that um we can find things such as instantaneous velocity, wow, we cannot find average velocity. We want to find displacement over a certain amount of time, such as five seconds. We can plug that in there. Um We can also consider um average velocity which is going to be S f E minus s of a all divided by being my inside. This can be average velocity. If we want to look at graphs for example and see when it reaches at the highest point, um we see that this right here is going to be um something that is going to model a particle moving along a line with its position. So it's going to look different than, say if we had a rocket that had parabolic trajectory where it reaches a maximum point. If we have something like that, we're able to see where that maximum point is reached, where it hits the ground and that's gonna be helpful in solving other problems.


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