Velocity and Acceleration as Derivatives of the Position Vector + Example of an Object moving in 2D by @mes on 3Speak
▶️ Watch on 3Speak - YouTube - Telegram - Summary - Notes - Sections playlist - Vector Functions playlist - MES Links --- In this video, I show that the derivative of the position vector of a parti...
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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, axis, 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. I see, 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 an increment by H at 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 take this here, put this in red, all right? So now we're going to do is I'm going to draw This is going to be the tangent vector, so we're going to re-hook call from earlier video. It's 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 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 all 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 you look at derivatives. So if you 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, so it's 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, 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, like that. 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 will 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, 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, and 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 rate, it becomes a tangent vector across there. Alright, 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, a 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, is a 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 get, 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 arc length, et cetera, so Ds over DT. 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. Alright, yeah, so that's what we have there. And now, again, that's the DS over DT. Again, it's just infinite to infinite to 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. All right, yeah, it's 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 is equals to the derivative v. prime. Yeah, this is, it is better, so the vector derivative of that respective time, divided by, this is going to be, actually, I'm not by 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 this 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 cubed t i plus t squared j vectors these are just the standard basis vectors so this is the 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 x x and y 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 cubed, so it's going to 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 D. 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 square plus y2 minus y1 square and then narka z0 so it's 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 uh this is the magnitude uh this is the v uh yeah so the vector i got of t this is going to equal to now square root and then we just take the component 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 in 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, 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 finals, 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 2, J. 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, these are i and jays. 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 T squared, or the J component, 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 cube 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 is 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 it. So this is the tangent vectors like this. And this is going to be right here. This is 3i plus 2. Yeah, it's going to be something like that, but regardless, this is going to be our velocity of one. This is at the one point right here, it's 3 plus 2, so it's going to go 1, 2, 3, and then up 2. So 1, 2, it's going to be something like that. And I'll draw it now in red, this part right here, the acceleration. The acceleration is going to be, it's going to be more flat, so it's going to be 6 and then 2. So it's in other words, it's going to go 6 here and then 2 up. And what I'll do is I'm going to move this down, move this down like that. And, yeah, actually, I move that above there, so move it over there. So now the draw on the red one, it's going to be, it's going to be flatter, so it's going to be something like that, and this is going to be our A of 1. This is going to be a vector, just put these vectors. It's going to be 6 to the right, and then up 2. The other one's going to be 3 to 3. to the right up up to. So it's basically like half the steepness over there. And what I'll do is, I forgot to draw this part. And this is going to be the arrow on top. So that's, that is that. And just fix it up, made it darker there, all right? So that is the vector there. And likewise, I'm just going to just use the figure in the calculus books. It's just, it gets more accurate. I'll just use this figure. So here's a figure from my calculus book. And, yeah, there is our velocity and acceleration vectors at 1. And for completeness, as always, I like graphing this out in an online calculator. In this case, let's also graph in the GOGBRA 2D graphing calculator. So here it is, and I made this a pretty epic calculator right here, or I mean, this little section of the calculator. So I added this, made it customizable, and so on. So there's our curve. You can just write curve equals tq, t squared, and then made it from negative 10 to 10, and so on. And you can adjust the slider here, A. and yeah, so you're going to adjust that slide, I'm not sure what this R of A is, I think I, I can remove that. Yeah, so then we have the point P, so that's the point is our position vector here, like that, and I will draw this in black and get rid of this thingy. Yeah, so this is going to be, this is our position vector, and then I also have the velocity, velocity, velocity is equal to, well, the position vector plus velocity plus p, and where velocity is just this. It's 3A squared plus 2A, that's just this part right here. I made it generic, so 3a, yeah, 3T squared plus 2T, and I put A so that I can have an A variable I can change. So this is the curve, so this is the curve, it will draw this fully out, but then I want to point with the vector always there, so I made a second variable A, so I can modify this, A, just for T. And then likewise for acceleration, we have 6A2, yes, so 6A, so 6T, and then I can modify it and so on. And again, this AAP and this RA, I'm not sure what those are, actually. I'll see if I might actually just remove them. Anyways, then you could just slide them over and see what happens. So I'm going to open this up, and let's see how this looks. So I'm going to open this up, make this wider. I just want to load in real time so you can see it all, loading. Let's see, okay, it's taking a wall, this is probably on the tablet. All right, so there it is. And yeah, it looks pretty epic, actually. I'm going to drag this around, and then drag this around. Move the A up top. So this part right here, I don't even know what this is for. Oh, that's AP. I'm going to delete this, actually. And then this other one, where was the other one? We had another VAL RA. So what in the world is RA? I'm going to delete…