A particle is moving along straight line with velocity (t) = 9 - t mgsec_ over the tme interval 0 < t < 4 sec_ Find the total distance traveled by the particl...

A particle moves along the $x$ -axis with the velocity history shown. If the particle is at the position $x=$ -4 in. at time $t=0,$ plot the corresponding displacement history for the time interval $0 \leq t \leq 10$ sec. Additionally, find the net displacement and total distance traveled by the particle for this interval.

First of all, let's draw X relation. Wears his time graph we know they had. Ex relation is equal to d be divided by DT. Let's differentiate the lost e Well, a bit lost is d square minus or G less three. So x relation is equal to do De minus four on me to per second squared. Now we draw x relation. Um, or since time craft using this equation, then we have on the craft on the graph. Looks like this, right? So it would look like this. Okay, you see is one point off time, right? So it's time to use equal to zero. Next relation is here right here. X relation is taken on the Y axis and time is taken on the pack. Six is now Let us throw a VT craft Lost the time craft Well, we lost your bhangra would be lost in the region for lost use a t square minus or are deep lis three. Now using this equation, we can plot a witty graft and the VT graph would look like this. Now this waxes. This is act sixes. We have time when XX is in on y X is we have a velocity is function of time. So it time physical to zero. The lost is here at this point, and then it could be crazies. Right? So here have been lost is negative. And then 18 crazies. Right. So the graph would look like these. So at this point receded. Ah, Here. Smoke zero. Therefore, at this point, X relation must be zero sins on the slope. Zero here.

Well, we have lost e lost. U V is equal to 12 minus three times D square. Ex relation is given by indeed be divided by DT. Let's different your velocity and he divided my detain to relaunch Steve, it is 12 minus three times square on differentiating this week did minus six times t Now let's find out X elation in time. Physical too. It Time. Physical. Four seconds. So acceleration time Physical. Four seconds is it Will do, um, minus six into four. So x relation is equal to minus 24 Mitac or second square. Right? And now we have the integral so integral off DS and limits on from minus 10 to yes is able to integral or three times DT and the limits off time are from one to t and therefore we have no s. Plus 10 is equal to 12. T minus. T cube minus 11. Now, let's find out s a time T is equal to zero. So s a t o. T. Is equal to zero is equal to minus 21 and s eight time physical to 10 seconds is equal to minus 901 No, let's find out deltas known deltas is equal to minus 901 minus line 01 minus minus 21. So don't ask is equal to minus 880 Right. And are we is equal to zero when he is equal to two seconds. Right? So, um, distance a time to use equal to two seconds is equal to talking to To minus took you minus 20 von which is equal to minus Fife and no distance. Time T is equal to 21 minus five less 901 minus five, which is equal to 912.

Hello, everyone. A particle moving along the line. It's a displacement according to the function, uh, ex ft. Uh, except he is equal to a T squared minus tootie plus four, where X is measured in meters and tease, measured in seconds. But in the average velocity or the time period, A T is equal to 02 I find it from 0 to 2. So the average velocity, um, is just a fancy way of saying to slope. So if you want to find the average velocity from 0 to 2, we're just going to plug. It was going to find this slope. Essentially. So when we plug in, um, well, this little was just why? To minus lie one over x two minutes, x one. In this case, no, uh, that X two is two. And why x one is zero. And I would have to find the corresponding my values. Several good to plug in to into our equation to get why, too. So we get, uh, t squared. So two squared minus two times two. Bless for which is equal to four. Um, so we have four of the top eyes. Why? To um And then for? Why one? We're going to plug in zero, so you get zero squared plus for so is also equal to four. So get four minus four, divided by two minus zero, which is equal to zero. So our average velocity over this time period is going to be equal to zero meters per second. So thank you for watching, and I hope this helps.

So even this loss of function here let's first find the displacement. So we're gonna do from 1 to 3 of 1/2 -1 over T. So that we get 1/2 team and selling teeth 23 And so that gives us 3/2 by this element of three And it's 1/2 plus zero equals one minus pound of three. That is that this placement there for total distance. Do 1-2 of negativity Plus for 2-3 of teeth. Okay so That means we get negative 1/2. Uh huh. She so that the teeth This 1/2 of T December 22: three. I want to to be negative 1% to that's what half was three house by selling of three I guess one plus islander to simplifying this here we got to Helen too minus felony three kids to cook apart being here. Now the displacement it's going to be equal to the total distance. The reason why that is is because this is Always greater than zero on the central here. So going to say from or 9 30 So that's equal to three T. to the one half. It's too Take up to six times 3 six times two She equals 18 -12 She goes six m.