Example 2: Velocity and Acceleration of an Object moving in 3D by @mes on 3Speak
▶️ Watch on 3Speak - YouTube - Telegram - Summary - Notes - Sections playlist - Vector Functions playlist - MES Links --- In this video, I go over an example of obtaining and graphing the velocity ...
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Example 2, Moving Particle in 3D. So this was in 2D, example 1, now we're going to take it a step further in 3D. So find the velocity, acceleration, and speed of a particle with position vector, R vector of T equals 2 and has components T squared, E to the power of T, then T times E to the power of T. All right, so let's take a look at the solution. So the solution will we take the velocity of this. The velocity is going to be the derivative of that, so it's going to be R of, yeah, the derivative of the position vector, R of T, so then put this all inside, it's going to be now T squared, derivative is going to be 2T, then ET is going to be, well, ET, that's a special property of the exponential number E, or just the number E to the power of T, and it's used a lot because the derivative is itself, instead of having to deal with logs or lons, yeah, I mean logarithms and so on. So it's going to be now T, Derivative of T using product rule is going to be well. Derivative of T is 1. So we're going to have E.T, and then derivative of plus now T times derivative E.T is ET. Like that. And now this equals 2. I just put it like this. This is going to be 2T, E.T, and then this is a factor out to ET. So it's going to get a 1 plus T. Yeah, 1 plus T like that. And then E to the T. Like that, that fascinating stuff. Alright, and now the acceleration, well, we're going to take the acceleration is going to be derivative of the V, or the second derivative of our position vector. Okay, this equals to, yeah, we take second derivative of that, or the solutions put, actually, V right now, let's just copy what it had, so V, derivative of the velocity, and that's just going to be a second derivative of that. This just equals 2. Now take the derivative of this one, it's going to be 2. Then E to the T is just E to the T. Now this one right here, again, likewise, the derivative of 1 plus T is going to be, well, this goes to 0, it's going to have a 1. We're going to have a E to the T, and then, again, likewise, plus 1 plus using product rule, then derivative of ET. It's going to be E to T. like that. So factor that out, we're going to get a 2, E to the T, and then we're going to get a setup like this. This is going to be now plus, yeah, this is a factored this out, we're going to have a 1 plus 1, in other words, we're going to have a 2. 2 plus T, E to the T, like that. All right, and now let's get the speed, and that's basically the magnitude of the velocity vector. So we're going to get the magnitude of the velocity vector, and using our giant square root, squared everything. So we're going to have a giant square root sign like that. Now, 2T squared is it's going to be, well, 4T squared, and now ET squared is going to be plus. E to T, and I'll write this a bit bigger, 4T squared, and then plus the last one, this one right here, 1 plus T, and then squared, 1 plus T squared, then E to the 2T, so that gets squared as well, E to the 2T, like that, all right. Yeah, all right, so we're done this question, but calculus book keeps going for this. So it draws the figure. So the figure below shows the path of the particle with the velocity and acceleration vectors when T equals to 1. So I'll try to draw this out manually, then I'll just show the, I'll take a screenshot from the calculus book, or just take that figure from it. So what we'll have is, so let's say we have our, this is going to be our X, it's going to be our Y setup. Let's make the Y like this, actually. that, and there is our Z. So, now we take our position vector, let's go to start from the beginning, this one right here, so R of T, yeah, so put it at one initially, I'll just put the whole thing, so R of T is equal to, uh, t squared E T, so it's going to be T squared, yeah, because I like trying to draw it out manually, just, it helps you understand how the, how the equations all plays out, so T and then E T, T, E, T, like that. So then R of 1, well, it's going to be, plug in 1 inside here, we're going to get a 1, so T squared 1, and then E to the T, actually, yeah, this is going to be just E to the 1, it's going to be E, it's going to be E, and then the other one as well, it's going to be 1, 1. And then also I want to get is the starting point. I'm going to actually put, notice what we have is R of 0. This equals 2. We'll plug in 0. We're going to get 0 for the x component. Then e to the 0, that equals 1. Then it's going to be well, 0 times e to 0.0. This is going to be, by definition, 01. So that's our starting point. I'm just going to only graph the positive values of T. So we're going to go right here and put in this part right here. So 0. This is the origin there, 0 there, and we're going to go to 1. So that is 1 on the y-axis, and 0 on the Z-axis. And then afterwards, it's all positive. This is just going to be t squared, e-2-a-t-t-t-t, then t-et-t. So it's all going to be positive. It's going to be, basically, increasing. It's going to be something like that. Yeah, something like that, and so on. Alright, and now, because it wants it at the point, basically at t equals to 1, so let's take a look at that. Now, all right, so the velocity at 1, V vector at 1, equals 2, and that's this part right here, and where is it, yeah, this velocity, so 2, so 1, so 2, then E to T is just E, then this one here, 1 plus T, e to the T. So in other words, that's going to be 2, 1 plus 1. E. So we're going to have a 2E, the T is just 1, and then 1 plus 1, this would be 2E. Like that. And then the acceleration is at 1 is going to equal to pretty much the same thing as the velocity, right here, because this is number 2T, this just goes to 2, and this is E to, and now we have over here 2 plus T, in another one, 2 plus 1, so it's be 3, then E, E to the 1. So we're going to have a 2E, 3E. So it's exactly the same thing as the acceleration, I mean, as the velocity, except the Z component is higher. So it's going to go up faster than the other one. So we're going to draw, I'm going to draw this one in, yeah, I don't know why, Congress will keep drawing it in blue. So it draws the velocity in blue. It can be similar like that. So let's say we have it at T -Eagles 1 here. and you have it, and it's going to be all tangent, so this one's all tangent, so this is V of 1, I'll put it here, put it here like this, actually, I'll just keep it like that, and then I'll draw it in black after, so yeah, so I'll draw this right here. This is going to be V of 1. Yeah, the calculus, the calculus book just does that, so I'm going to draw now in red. and the red is going to be, well, it's going to be the same direction as the V, except it's going to go faster up. So it's going to be actually faster up. So, Mag2 is going to be slightly bigger as well. It's going to be up like that. And this is going to be our, I'll make it like that. This is going to be our, I'll just put it like in red. Okay, so this can be a vector like that. Yeah, so that's going to be more vertical. And at this point right here, I'll make it in block, this is at our t equals to 1. And again, notice how it equals to 1, we have 1, E, E, so we go to x is 1, so it's going to be actually like this, x is 1, so this is going to be x is 1 there. Yeah, that's 1. x is 1, then e, so it's going to be this number, it's going to be somewhere around here, e, e is 2.618 or something like that. and then, yeah, then the other one's going to be E above as well, this is 2.618, is E, yeah, so it's E, yeah, so that's pretty fascinating stuff. All right, here, quickly pause, video, and just googled E equal, so E is actually 2 .718, not 618, alright, so that's pretty close. I also saw 618, okay, so I guess it's 7. Anyway, so that's what we have, and what I'm going to do now is I'm going to erase this, and then just use the calculus book drawing, because it's neater, and then just write over it as well. So that's what we have. I'm going to close this part here. All right, yeah, so that's what we have. So, again, bunch, bunch neater. And now what I'll do is I'll write down the stuff that we want. So this is going to be somewhere here. And this is going to be at our X is 1. So I'll write this like this. This is X is 1 here. That's our 1E, and then we're going to get all the way across here. This is going to be at our, I'll draw this at E. And this is, again, it doesn't look like it's to scale either, so that's interesting. So, yeah, drawing isn't that, well, it's going to be like this to 1-1-1 -1. I'll draw this. Did better. One more, okay, let's try it again. Dash 1, dot, dot, dot, dot, like that. Yeah, so it looks something like that, yeah, so it's pretty fascinating stuff there. And now we'll grab this in GOG, bro, to see how it actually looks like. But I also have T from negative 10 to 10, but again, You can know, typically ignore the negative values there. Yeah, and yeah, that's typically because time, yeah, it's time just moves forward always, so you don't have a negative time and so on. But, nonetheless I do graph it in negative, just to see how that looks like. Anyways, let's click over there. I'll do that soon. So there's the graph, and then there's their A value. You can adjust this. If you go to A equals 1, and it will get you this set up here. You can also go to A equals 0 to get to the bottom there. I'll graph it out soon. Again, as before, So here's what I did. So here's the, I made the point A as R of A, the position vector. So I put this inside this one at A there. So where A, you can make it a variable. So you could A, this will put it as 1, 1.2, and so on. All right, so that's A. And so then the acceleration as before, I go, the way you would write it, you go vector A plus, so you're going to have a displacement one. So you go from the A point. and then over to this vector acceleration. And then there's acceleration, so we're going to have it as 2E to the power of A, and then, yeah, 2, comma, EA, and then comma 2 plus A to power of times e to the A. Again, you put T, so graphs out the whole thing, and then A for specific points, and this is a vector, and then plus A. And then likewise here, for the velocity, you go vector A, and then A plus, There's the velocity vector 2A, yeah, 2A, E to power of A, then 1 plus A times e to power of A. Again, that's just this part right here. So, this is 1 plus T or 1 plus A. There's 2 plus A, and then everything else is the same. Well, yeah, this one's actually 2T, this is 2, and then ET, so that's the difference as well, so 2A. So there's the 2A for velocity, and then you have multiplied... I mean, take the derivative 2 for the acceleration. When you get something that looks like this, it looks just like this part right here, and we're going to graph this out, see, let's play around with it. All right, so I'm going to close some stuff. Yeah, so it's going to load up. Yeah, so there it is, and there's our setup, and you can adjust this, make it at 1. And, yeah, you could adjust that, you could make it there, and you could also show it, so what you could do is, so you can move this like this. Goes down, let's go to 0. Where does zero go to? Yeah, so zero goes to the y-axis at 1 there, so that's pretty cool. And then make it at 1. Looks like that. Where it's literally parallel to it, or like, yeah, parallel to it. I'm sure, well, parallel from the other side. But yeah, basically, it's in line with it. It's the same plane, I mean, not parallel, but... I'll say, I would say it's like, yeah, basically the same thing except it's pointing upwards, so yeah, a better way to state that. So it's basically, this is just a bit higher because you have the 3E, this one is 2E there. So that's pretty cool there. And then what I'll do is, just see how this graph looks like. Let's say we go from here, instead of negative 10, we go to 0, and then go to 100, see what this looks like. Yeah, it goes like that. Okay, this is a... Is this going to be pointing up more and more? Is it a way to shrink that? Okay, shrink it there. I'll make this shrink. Oops. I'll shrink this there like that. Okay, it goes up. Anyways, it will look something like that. And what I do want to do is go back to this negative side. I'm going to go undo. Undo. Get the negative sides in there. Okay, so there's a negative sides. Yeah, so the negative one looks like this. It's pretty interesting. It goes underneath there. and so on, that's pretty cool. And it's, I think I would assume it goes to zero, close to zero, because you're having this part right here, E to the T, but now if you have E to the negative, it's going to be the same thing as writing 1 over E, so you're going to have a dividing by a big number. Same thing with here. So the Y and Z components get shrunk, whereas this one just becomes positive anyways. So anyways, that's a pretty fascinating stuff.