Motion in Space: Velocity and Acceleration

Transcript

Hi, it's time for another math. Easy solution. Today we're going to go over further into vector functions and now look at motion in space and basically look at velocity. and acceleration. All right, and before it go further, as always, if you don't have time to watch this whole video, you could play this video at a faster speed, and top secret life fact your brain gets used to faster speed, and here's a tutorial I did on the, just downloading the browser extension and so on, so definitely check that up. And also, you can also download and read the notes in PDF format, as well as in article format on the Hive blockchain, and the link will be in the description below, as well as the video timestamps. So, time stamps of all parts are in the description as well as the video section's playlist, and yeah, basically I'll be breaking down this long video into many smaller parts, put that all in the giant playlist, and so on. And again, with the timestamps, you can just skip ahead to wherever you want in this video. All right, going further, so the calculus book reference, note that I mainly follow along the book Calculus, early transcendental, 7th edition by James Stewart, there's a link to it, and for the whole chapter, I use the solution manual for this here, but for this video, I don't actually have, don't actually use it for this video, but for the chapter, I use the link is over here, and the notes, and also note in some earlier videos, I used the sixth edition, because I had the physical... a book there. Anyways, going further, so calculus book chapter, so the hive notes and sections playlist for each video of this chapter are listed below. Yeah, so here's a whole calculus book chapter, there's vector functions, and this includes this playlist to all of these videos so far. I've done three of these, so vector functions and space curves, and there's a sections playlist, this is the hive notes, and there's derivatives and integrals of vector functions. and then the video before I did arc length and curvature, and now we're going to cover motion and space, velocity, and acceleration, and it's the current video, so it's going to be quite epic. And note that these links are also on MES links. If you go here to my website here, I basically added all these in there. You can click this button, and you can click the math tutorials, and then you'll see all these here, so you can shrink this. and so on, and then you go to the vector functions, it's going to be in here. So it's going to be like this. You can see them all there. In an infinite sequence series, I also have one spherical harmonics vectors, and the geometry of space vector functions, and so on, right, pretty fascinating stuff. Are you going further? So topics to cover, note that the timestamps will be included in the video description for each topic listed below. So we have motion and space, velocity, and acceleration. Go over example 1, moving particle in 2D, then example 2, moving particle in 3D, then example 3, integrating to get velocity and position vectors, example 4, centripetal force, and then example 5, horizontal distance of projectile, and then example 6, impact speed of projectile, this will actually impact on the ground. So tangential and normal components of acceleration, then go over acceleration vector in terms of derivatives of R, or the position. and then example 7, or in terms of, yeah, derivatives of the position, of a position like that, nor write it better, so in position vector. So, acceleration vector in terms of derivatives of the position vector, and then look at example 7, tangential and normal components of acceleration. All right, and now let's go finally into the chapter or the section, so motion and space, velocity, and acceleration. So in this section, we show how the ideas of tangent and normal vectors and curvature can be used in physics to study the motion of an object, including its velocity and acceleration along a space curve. In particular, In particular, we follow in the footsteps of Isaac Newton by using these methods to derive Kepler's first law of planetary motion. This will actually be in another video after this one. It's a number of this one is motion space. After that, I'll do an applied project on Kepler's laws. They'll be in a future video, so it'll be quite fascinating, so stay tuned for that. So let's continue for that. So suppose a particle moves through space so that its position vector at time T, is R of T, so this is bold for a vector. All right, so let's say, suppose we had, let's say, here's the X, Y, this is going to be the Y, this is the X, this is the Z, so this is in 3D, so let's say we had a curve, it goes something like this, okay? This is our curve C, and let's say we had a point P here, close P, and let's say we had a point Q here, and what I'll do is I'll move this down a bit. Move it down like this tiny bit, so it goes down like that. Alright, so what we have here, I'm going to erase this, put this on top, like that, Q, and then this is, let's say we had, this is a position, position vector R of T like this. All right, I'm going to move this actually above it, slightly higher. So it's somewhere, then, also just shrink it like this a bit. And now we have, let's say this is the origin here. So this is the origin. All right, so yeah, so there's origin. I'm going to draw this line like this. This is going to be now R vector T plus H. That's a position vector, and then this is at time. This is going to be at just increment by H the time, and it moves over there. All right, and so what I'll do now is I'm going to draw this in red. So I'll take this here, put this in red. All right. So now we're going to do is I'm going to draw the, this is going to be the tangent vector, so we're going to recall from earlier video. is going to be the derivative, r prime of t like that, and that is the derivative there, and now this, I'm going to draw this one now in blue, in blue, let's put it like this, this is this blue. All right, so I'm going to draw from here, from p to Q, so this right here is a vector, this is going to be actually r of t plus h minus r of t. And yeah, so, and you can see this because this, if you just use basic vector addition, so R of T, so if you take this one, and then you add it to, I mean, then you add it to this one, what ends up happening is, these just cancel out. So this would cancel out with this one, you're left with this. Yes, you're just left with that, yeah, so we can basically erase all that, and yeah, so that's what we have there. And recall now, again, from, in my earlier video on vector functions and so on, when we look at derivatives. So, extend this over here, so now this vector, and I'm going to draw this actually in black now. So I'm going to draw this in black. All right, and let's just close this part. So this one here, I'm going to, yeah, this one is just, we're going to multiply basically this right here. by 1 over H. So we're going to multiply that by 1 over H, where H is the constant, again, it's just this incremental thing there. In other words, it's just a constant multiplying by this, so it's just going to be a scalar multiplied by a vector, so it's still going to be parallel to this, this is going to be straight across it. This is going to be, and I'll just put an arrow, so this vector is just going to be R of T plus H minus R of T, like this, divided by H, like that. All right, so now we have this, and also I've had complaints about my drawing, so what I'll do is I'm just going to use the calculus book, like, image, and I'll add that part there. And here, quickly just pause the video and added this over here. So here, yeah, it looks much nicer than what I drew, but I'm just going to draw this here, arrow. This is going to be our, yeah, this is going to be our R of T plus H minus R of T. Yeah, without the minus, yeah, and without the divided by H there, that's just this part right here, so this parallel to this right here, parallel. So anyways, going further, so notice from the figure above that for small values of H, the vector, and that's this part right here, so this vector, R of T plus H, minus R vector of A of T, and this vector, and then just divide this by H, Yeah, so this vector, this approximates the direction of the particle moving along the curve R of T. So as it's going here for small values of H, this is pretty much the direction of the particle. Imagine this H was super, super small, be right over here, and that vector would be literally pointing in a direction that this particle is moving. This particle is moving across like that, and that's going to be basically the direction of it. So its magnitude measures the size of the displacement vector per unit. time. So in other words, yeah, in other words, this from P to Q, that's a displacement, and then this H, that's a time variable. So that's basically what have units of seconds, if it was in seconds. So because you're going to be dividing it by this H there. So you're going to have displacement divided by time. In other words, you pretty much have the velocity. So the above vector gives the average velocity over a time interval of length H, and its limit is the velocity vector V of T at time T. So that's average velocity. But then the limit as H approaches 0, you basically have the limit of a derivative. So in other words, now what we have is the velocity vector V of T. This equals to the limit of that displacement vector as H approaches to zero. So this is going to be R of T, T plus H minus R vector of T. divided by H, and now this just equals 2, well, this is just derivative. So again, definition of derivative for vector functions, see that in my earlier video and video series, so r prime of t. So the velocity is just the derivative of the position vector, which is exactly what we already know in basic calculus. Yeah, so thus the velocity vector is also the tangent vector and points in the direction of the tangent line. So the speed, yeah, again, going back there, again, here's the, this one here, this derivative, well, this is the average velocity, but then as the limit of age goes to zero, this gets shrinks and becomes lines up with this radio, it becomes a tangent vector across there. All right, so thus, yeah, that's over there. So the speed of the particle at time T is the magnitude of the length of it or the magnitude. That is the, this right here, absolute value, sign of the vector V of T. So this is appropriate because from the above equation and from the equation from my earlier video on the arc length function, we have, so recall from my earlier video, here's a thumbnail of that, one of these over there. So basically, this is the arc length function. In my earlier video, I showed that the arc length right here, from here to here, this is basically equal to the integral from A to T of the derivative of the velocity vector there. Because remember, the position, essentially that's like a displacement or the position, so then the integral of the velocity, you'll get that position there. and I show that it's equal to this thingy, and then this equation is going to look at, the derivative, DS over DT, so derivative of this position vector is equal to, again, using the fundamental theory of calculus, that's the integral part in there. So derivative equals to this, and then that's why if you take the integral, you're going to get the S, this one's going to be, well, the integral of that, that's this whole part. All right, so that's what we got, DS over DT on that. So what we have is, in our case, we're going to have the speed, is the Epsilon value or the magnitude or length of the velocity vector, V of T, and this equals to, well, that's just going to equal to this part here, because we know the velocity vector is, yeah, the velocity vector is, yeah, the velocity vector is just equals to the derivative of the position. So that's just going to be, oh, the tangent vector, that's the tangent vector, this equals to the absolute value of R of prime of T. And now this is equals to DS over DT from our earlier video. So Ds, that's the R-clength, et cetera, so D-S over D-T. All right, and then writing this out, what this means is basically, well, this is just rate of change of distance. distance with respect to time. Now put it all like that, respect to time. Now put it all like that, respect to time. And again, Congress will put it all in one like equation form, so I'm doing that as well. All right, yeah, so that's what we have there, and now, again, that's the DS over DT. Again, it's just infinite to infinitesimal, small, uh, changing distance, divided by infinitesimal small, uh, the time that. So basically this right here, so, but like super, super tiny, DS over, uh, over time. Yeah, something I got, or, yeah, some pretty, um, epic stuff there. All right, and just as in the case of one-dimensional motion, the acceleration of the particle is defined as the derivative of the velocity. Yeah, so to get the acceleration, well, it's just going to be the A vector, like that acceleration vector, this just equals to the derivative v prime. Yeah, this is better, so the vector derivative of that respective time, divided by, this is going to be, actually, I want to define anything, that's just like that. Yeah, just the derivative of the velocity vector, and then this is going to equal to now. Well, remember this one right here, so the velocity vector is equal to the derivative of the position vector, so in other words, it's the second derivative, so just double derivative of the position vector R of T like that. So that is quite fast. I'm actually bold this because this is pretty important. Yeah, so we bold this one right here. I'm also going to bowl the velocity here, too, but this is pretty... big formula that we're going to be using that's velocity one, again, just equals to using definition derivative, and then you get this part right here, derivative of the position vector, and then the velocity, acceleration defined as the derivative of the velocity, or the second derivative of the position vector. Now let's take a look and example on this, so example one, moving particle in 2D, so the position vector of an object moving in a plane is given by... r of t equals t cube t i plus t squared j vectors these are just the standard basis vectors so this is x component so we're asked to find its velocity speed and acceleration or exploration right here just fix that up and illustrate geometrically all right so remember in a plane this just means in 2d and we're given two dimensions right here, I and J or X and Y. So solution to velocity and acceleration at time, T, R, so that's R of T. So what we'll do is, well, we're going to write here is the velocity is going to be the derivative, R prime of T, that equals to, well, derivative is going to be, well, there's the T, uh, T-cube, so it's going to be, take the derivative, so 3T squared, I, plus, and then 2T, the T squared becomes 2T, the derivative, this is going to be J vector. And then likewise, the acceleration, this is going to equal to the derivative of the velocity, which is the second derivative of the position vector, so this equals to R double prime T. This equals 2, now take this down, it's going to be 6T, so put the 2 down and subtract 1 power, so yeah. 6TI plus, and this is going to be 2J, like that, all right, and the speed now, because we're asked to find the speed as well, so the speed is obtained from the length or magnitude of a vector, of a vector formula, yeah, so this whole thingy right here, or actually a better, writing a bit better, and the speed is obtained from the formula. for the length or magnitude of a vector, which I showed in my earlier video. Again, this distance form in 3D and 2D, actually, because, well, if the Z1 is a 0, just get rid of all. Yeah, so that's just going to be like this, and is just basically using Pythagris. So the distance from P1 to P2. as equal to x2 minus x1 squared plus y2 minus y1 square and then RKZ0 so just ignore that and then all square root so thus the speed is the uh right here is the absolute value uh sign there so speed and I'll write this down as speed so it's just a magnitude of the vector velocity vector I mean of the velocity vector it's a magnitude of the velocity vector so this equals to this is the magnitude uh this is the v uh yeah so v vector i got of t this is going to equal to now square root and then we just take the components so three uh three t squared and two t so we have to square those so it's going to be three but this complete for completeness for this first example so three t squared squared plus and then two t squared squared 2T squared, and we're going to square root all that. Like that, so this is going to equal 2. Now, this is going to be, just simplified. This is going to be 9, so 3 squared is 9, t to the 4, plus, now we're going to have 4 t squared. All right, now the next setup is well, when t equals 1, and again, this is actually forgot to include this. So, finest, velocity, speed, and acceleration, when t equals 0. and illustrate geometrically. So, all right, so when t equals to 1, we have, let's get all these values here, so we have a V of 1, this equals to, we'll look at our velocity right there. This is going to be 3T squared, so 31 squared, and then plus 2T, that's just going to be, well, 3i plus 2J. All right, and then the A value, that's just going to be the acceleration, I'll put it like this, actually, so acceleration at 1, this equals 2, well, I'll plug that in, so this is going to be, well, 6 times T times 1, so 6 and 2. So that's just going to be 6i plus 2J, and then lastly the speed, so this is going to be the velocity, the magnitude of the velocity, at 1. is equal to, well, let's plug that inside, so we're going to get a 9 times 1 plus 4 times 1, yeah, square doesn't, 1 squared doesn't change anything, so 1. So this is 9 plus 4, that's equals to square root 13. All right, so now let's draw these out. So these velocity and acceleration vectors are shown in the figure below. So what we'll get is, well, we have to go back to our position vector. So R of T, this equals to, just to see how it would look like. This is t cubed i plus t squared j. So this is going to be t cubed i plus t squared j. Yeah, so notice here, so basically, yeah, basically for negative values of t, what we end up having is, oh, well, this is going to dominate right here, it's going to be negative. Actually, no, we're dealing with parametric, yeah, these are i and jay. So for negative values, The x component, this is the i1, that's going to be negative. So in other word, we're going to have something like this. But we're going to ignore the negative side, mostly. So I'm going to go this x, this is y. So for negative values of t, we're going to have negative values, but then this right here, this is the y component, the y component, the j squared, the j squared, is always going to be positive. So it doesn't matter if you go negative, it's going to still, it's going to be like it's always going to be positive. And this is on the squared side, so it looks like a problem. So basically it's going to be like parabola like, but then you have this T cubed there, but it's going to look something like this, and what we'll do is this is going to be the origin, so we're going to ignore the negative sides more. And now at the point, and also if you plug in 1 right here, so this is right here, R of 1, this just equals to 1, well, 1, yeah, i plus j, because yeah, the t squared or t cubed just 1. So that's just going to be 1 1. It's going to be somewhere here. Yeah, so that's our 1-1. Yeah, so I'll put that like this, 1 -1, and then the tangent vector is going to be, I'll draw this in red. Yeah, I'll go over here, I'll draw the tangent vector in red, or actually I'll draw that in blue, because the calculus book does…

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