All right. So what we have here is a more challenging problem just because there's so many time intervals that are occurring. But if we draw really, really careful diagram and that will help us not lose track of what's going on. So um, I think it would be most useful to just use a really, really long number line for this problem because it's one dimensional or only horizontal world ex access. We don't need to draw a holograph.
We don't need to worry about vertical. Why direction? We're just going to drawn number. Hawaiian. Teo represents our X axis control. The there dot right here.
It's indicated That's our starting point. That's where the puck is and our puck starts at a location of ex not is equal to zero as well as time is able to Sarah straight all drama and blue. So see that Tina is equal to zero and we're going to travel along to three different time intervals. So the permission that we rely so far is that it starts off a Tex equal zero and Tiegel is there. I'm just running ahead.
We're going Tio cross by three other time. So they're they're two seconds. Five seconds and seven seconds. I propose naming those by t ones he to Auntie three respectively. To t one is going to be gold two seconds and just to save space, I'll not worry about the significant figures on my diagram Butty too is one to be equal to five seconds and then likewise t three is going to be equal to seven seconds on DH then we don't know Well, I kind of dropped drew them evenly spaced here not to imply that they're all going to be evenly spaced.
But these are frankly unknown distances as yet that we have to solve for throughout the problem. So we'LL just follow the subscript convention Call him X one X to and X the re So this is our schematic of what's going on in the problem on And then the reason why the puck is even moving in the first place is because there may or may not be a force acting on each of these time intervals and I'm drawing and green right here. So that means is that we have a non zero force. So that is this zero point two five Newtons s So in the first time interval the forces being applied. Then again, with that things were about sitting figures zero point five Newtons.
But in the second time, interval the forces knowing we're being applied. So that gives you a force zero. Um, but then in the third interval, the forces again turned on because the player is pushing on again. So we again have a force of zero point two five. Newton's going on here, So those are the complexities they're going on.
We can already see if the force is going to be grilled. Zero. Then Newton's second law, which is is equal to M. A, is going to tell us that the acceleration is equal to zero. Um, but the acceleration is going to be non zero in the first and the third interval.
But anyway, I'm getting a little ahead of myself. So oops, um, it's a circle that a different color. In part A. We'll be solving for this position as all of its respective lasa ti and then part B, we're going to be solving for X three, the final position and velocity. This's our schematic diagram of what is going to go down in this problem.
And so let's just go ahead and get started with a mama with party way have our very carefully, carefully crafted diagram. So actually going to be using Kinnah Matics equations he's going to tell in order to solve for the conditions and velocities. So x one is going to be equal to the initial position. X starts added to the initial velocity. The nonce on Sorry, these were supposed to be current scenes.
Um, but the time interval from T won t zero and we super explicit here. Plus one half of this acceleration from T one through T zero squared is important on the acceleration term. So you can already see here. One other piece of information that was given to us they're about to skip job is that we also know that the pup begins at arrest. So, in other words, it's initial velocity is equal to zero.
That really helps us sounds. First equation right here. Tres velocity is you will seriously at second term ghost is there. Um, and in addition, the starts, the origins, its initial position is equal to zero. That will helps out there.
So we only need to plug in numbers for the third term. So you have a one half at front on acceleration that is going to be equal to zero points. Fine. Zero by your points. One six oh times.
This time in a balti points zero zero minus zero square. Which two? Three significant figures will give us three points. One three meters. Great. So that's the first half of party, eh? I think we have enough space down here to squeeze in the velocity so we can use a simpler can Max equation.
That being the one is legal. Teo, be not simply added Teo, the product of our acceleration multiply by the time interval that were working over sense from tees here T one squared No, one half, just like before. We know that the initial velocity is zeros. The first term goes away. So we are just left with that same acceleration which is force divided by mass.
Sorry, there's our acceleration that our time interval square, anything like that. And then at the end of the day, so it's starting to look a check mark of the one is remarkably. You can see the same answer because of the coincidence that the time intervals two seconds squared before but canceled out with the one after prints. That's also a numerically through going three. But the velocity so has different units comes with meters, units of meters per sec, second and knots of anything else.
So that wraps up part a. So remember diagram. I have the have the diagram wind down. I needed a new slide, some more space. So we ultimately want to know what the position and velocity are at that third going time more.
To get there, we have to calculate the position and velocity at X to first. So luckily, there's no accelerations. So the velocity that Position two is simply going to be the same as this one, which we said earlier was through Portland three years per second. However, because we're still travelling at a constant velocity, we only to apply the kingdom attics inclusion again in order to Teo for our new position. So final velocity is equal final position and is able to initial position plus Thea velocity that initial velocity was time interval.
Um which is from T born to t to that is too one half the acceleration industry Super explicit thie. Any potential acceleration that's occurring along this time Interval squared. So you know, from our careful diagram large earlier that this acceleration is zero. So we don't need to worry about their term. However, we do have to worry about the first two terms.
So we just said in the previous part we got three point one three meters two, three, six figures, but is actually three point one two five. So this is a good instance. Where when you have intermediate calculations and problems, you should use as many significant figures as you can. Don't round off intricately answers so that it doesn't affect your final answer. So we're going to be careful and include That s our new time.
Interval is between two seconds and five seconds s. So that's going to be that. And that will give us an ex too. That's equal, Teo. Exactly twelve point five meters.
So we can underline that that's not the final answer, but it's necessary in order. Teo, get to the final answer because we have a new ah, another time interval Just wanted for an acceleration. And that's between extra next three. So this to r V. tube velocity.
Then the time intervals from Teo Teo three get the one half acceleration again in this interval. And that is still from T two t three. Mr. Terrorist Squared squared. Okay, so none of these terms are equals zero.
So I need to put in something for all three. We just found out west he was. And that is twelve point five V two. He said is the same as being one. And to forcing me from figures that was through twenty five.
And then our new time intervals from five seconds to seven seconds. That gives us a seven. All right, Fine. Then add that to one half and then remember that were cackling our acceleration by dividing the yeah force by the mass sensor of one six zero acceleration on in time Thinking that squared this time seven rights five squared pelo that all into a calculator and then two, three student If figures could get tourney horn point nine returns. This is X three, which is one of the quantities that they are asking for in party excellence today.
I didn't label that we're doing party. We're gonna party. Here we go. Okay, So we are not signing at the last thing that we need to do. So trust little box in the corner.
Walk over here now is just offer that velocity at time, interval or sorry time three. So we already know that would be too is This is It's the same as the one because there's no acceleration. But the three is going to be equal. Teo be too. Plus the acceleration a lot was time interval, which is from course two to three.
Kalou's princes. You should go back pretty quickly. The full on round answer for Bi two is three twenty five at the again to our acceleration. And then that same time interval should try to fit. It's a seven point zero zero minus five one seven zero her flute Fitz.
And if it's yes, okay uploaded onto a calculator and you get a walking from the velocity of six point two five meters per second. This is the three right, and that wraps of party. So we are older.