 The universe is in constant motion everywhere we look. Consider a simple moonrise over the ridge of mountains in the distance. Consider more human-made motion, for instance the act of launching a rocket off the surface of the earth, putting it into some distant path that takes it around our planet. Or consider a simple game of pool. Here you look, objects move in more than one dimension. In pool, they can move along the surface of the table, which is two-dimensional motion. They can even hop, one ball can hop another if struck properly, which is motion in three-dimension. Let us begin to explore the tools that are needed to describe motion in more than one dimension. The key ideas that we will encounter in this section of the course are as follows. We will learn that we can combine one-dimensional descriptions of motion, that is position and time, displacement, velocity and acceleration, with the vector concept and extend our description of motion to the full three spatial dimensions. The concepts of average and instantaneous quantities can also be extended into three-dimensions. And finally, we will see that three-dimensional motion, and for the purposes of this section of the course, we will primarily concentrate on two-dimensional motion, is not simple, but we will learn how to deal with it using the toolkit that we build in this part of the course. And some very beautiful examples of motion in at least two dimensions are illustrated in the graphic on the right, which was produced by NASA to show the passage of the comet Kohutek, near Earth in the period between 1973 and 1974. The comet's trajectory, as it slingshots around the Sun, is shown in red, where the nearly circular orbit of Earth, as our planet goes around the Sun, is shown in blue. And you can see that there were two points of close approach between the comet and Earth in this motion, once when it entered the inner region of the solar system and once when it exited the inner region of our solar system. You can imagine that it's extremely important for us not only to be able to describe motion in two and three dimensions, but to use what we know about the laws of physics and descriptions of motion to make predictions so that we can understand when bodies from outside of our planet, for instance, which in the past have caused at least one major extinction on the planet, are going to pass near our planet so that we can observe them, but also understand if things like that will be a threat to life as we know it. Let us begin by looking at position and its description in more than one dimension. And for the purposes, as I said earlier of this lecture, we're going to concentrate on two-dimensional descriptions, but you can fairly easily see once you grasp the mathematics of two dimensions how one would then extend it to three and extend the ideas learned here into the additional third spatial dimension. Let us begin by thinking about a point in space and we know that we need a coordinate system in order to describe location in space and that coordinate system needs to be divided into regularly sized units. So let us employ a Cartesian coordinate system with an X and a Y axis. These axes are delineated in units of centimeters and we can see here that our coordinate axes span from zero to three centimeters in the horizontal and vertical directions. We place a point in this coordinate system represented by the blue dot and we note that it has a position whose location in this spatial system can be written using X and Y coordinates with the subscript one to indicate for instance the first position of an object and we can read off of the coordinate system that these are at 1.5 centimeters along the horizontal and 2 centimeters along the vertical so we can denote this point in space as 1.5 comma 2 in units of centimeters. Now we can go back to the vector concept and we can just as easily represent this point in space as part of a vector. The vector begins at the origin of the coordinate system and it extends until its arrow head reaches the point 1.5 comma 2 and we've seen how to write a vector with these features and so we would write this vector in two dimensions using the letter R and again since this is the first point in space first vector that we're going to write down we can denote it with a subscript one and of course we have to put the vector arrow hat over the top of the R and so in vector notation we see that this is a vector that has a horizontal component of length 1.5 centimeters pointing in the positive X direction denoted by I hat and length 2 centimeters in the vertical direction denoted by J hat. Now we could write a second point and we can represent the location of that point in space again using coordinates in our Cartesian coordinate system. Let's imagine that this point represents the position of the object we first started describing a moment ago with X1 and Y1 at some later time for instance time 2 and let's denote the location of this red point in space the later location of the same object as X2 and Y2 and we can again read off the graph and we can see that the object is now located at this later time at 2.5 comma 1 centimeters in X comma Y coordinates and we can again represent this as a vector now drawn in red as an arrow and this vector begins at the origin and has the components shown here which we take directly from the coordinate location of the point which is where the arrow head ends. Now we have two vectors representing two locations in space the first location at as a position of an object at an earlier time the second location is the position of the same object at a later time and we can ponder the following question what now in two dimensions is the displacement from the earlier time to the later time so think about that question for a moment maybe try working out the answer for your own without having really advanced through the subject very much use paper use a writing implement when you feel like you would like to move on you can can do so but I would encourage you to pause the video at this point try to work the answer out for yourself based on what we've learned about vector and specifically vector arithmetic and then when you're ready to resume resume the video the displacement is still and as ever always defined as the difference between the two positions with the earlier one subtracted from the later one so the earlier position was given by the vector R1 the later one was given by the vector R2 now in one dimension if we only had one coordinate axis to deal with for instance the x-axis and we had an early location x1 and a later location x2 we would have written the displacement as delta x equals x2 minus x1 and in reality nothing really changes this definition remains the same we merely extended to two or more dimensions in more than one dimension the displacement is the subtraction of two vectors and that makes that resulting object from the subtraction also a vector as we saw in the first lecture on vectors taking one vector and adding or subtracting it from another results in a vector so the displacement in more than one dimension can be written as a vector and we will denote this as delta R vector where the vector arrow hat goes over the letter R the delta again indicates difference and so what is the difference we're going to compute it's the earlier vector subtracted from the later one so R2 vector minus R1 vector and again in vectors part one we learned how to subtract vectors mathematically and graphically so let's try each of those approaches and see what we can learn about the answer to this question for the example shown on the left let's do graphical subtraction first to do the graphical representation of the subtraction of two vectors you draw a vector that points from the arrow head of the vector being subtracted that is R1 vector in this case to the arrow head of the vector from which it's being subtracted that is R2 vector in this case so let's take a look at what this would look like we're going to draw a vector that goes from R1 vector to R2 vector from arrow head to arrow head with the tail of this sub of this arrow representing the difference the displacement starting on the earlier vectors arrow head and pointing to the later vectors arrow head and so if we go ahead and draw that we see here delta R vector represented graphically without writing a single mathematical equation we've arrived at a possible answer to the question what is the displacement now of course we want to always have at least two ways of assessing the answer to a question that helps us to build confidence and the reliability of the results that we obtain so let us also do the mathematical approach let's do the mathematical subtraction this will allow us to gain a representation of the actual vector delta R vector so we begin by writing down the definition of the displacement in more than one dimension delta R vector equals R2 vector minus R1 vector from the previous slide we have the mathematical forms of R2 and R1 so we substitute those in square brackets where once we had R2 vector we write it out and it's full x and y component glory and we do the same for R1 and we subtract these two bracketed quantities from each other to represent the subtraction of R1 from R2 now I'm not going to go through all the steps you should practice this at this moment so I'd recommend pausing this video and making sure that the answer I show you here makes sense to you but what you should find when you do the subtraction grouping together terms that have common unit vectors and then applying the arithmetic operations to their coefficients to get simplified coefficients that singly multiply each unit vector i hat and j hat you should find that the x component of the resulting delta R vector has a length of one centimeter and points in the i hat direction positive so positive along x and that the y component also has a length of one centimeter but it does not point up in the positive y direction it points down in the negative j hat direction against the direction of positive increasing y values so we have two answers a graphical one and a mathematical one are they the same well we can begin to check the math answer against the graphical one the math answer describes a vector with a horizontal component pointing to the right and a vertical component pointing down and in fact that's exactly what our delta R graphical vector the arrow here does it has a component that points down and a component that points right and that's how we get from its tail to its arrowhead now you could also additionally print this out on a piece of paper and take a ruler and verify that in fact the length of this side of the right triangle representing the vertical component is the same at least as approximately as you can get it with a ruler and paper as the horizontal component or the horizontal side of this right triangle there are many checks you can do at this point this is already a good one just to get the sense that these answers are consistent even though we used two approaches on this slide now motion in more than one dimension doesn't only have to be the kind of simple implied straight line motion from point one to point two that we just explored that was to motivate us but now it's time to unscrew the training wheels from this exercise and begin to dig a little bit more deeply into displacement and complex motion in at least two dimensions so again on the left here I have a Cartesian coordinate system it has an x-axis and centimeters is a y-axis and centimeters I've expanded the axes now to go from 0 to 5 centimeters and let's think about the kinds of complex motions in two dimensions that an object might take we can think of that complex motion as a large collection of points each with its own x y coordinate and those points are so closely spaced that we can't see the gaps between them and they appear to our eye as a continuous and unbroken line of motion in x and y and this is represented by the black curve shown over here on this Cartesian coordinate system on the left this represents a motion that at time 0 begins at the origin 0 comma 0 and proceeds along the curve for later and later times until we reach this point back on the x-axis here just before 4 centimeters on the x-axis now note that time itself is absent from this picture in other words you can't tell whether points that are near each other for instance here on the motion curve occur within close times or whether the number of the amount of time for instance that passes in this early part of the curve could because of the forces involved happen to be equal to the amount of time represented by the rest of the curve up to the apex of the curve I just point this out not to confuse you or confound you but to note that you need to be cautious when looking at a graph only showing position in two dimensions the time is not represented here and so you don't really have any knowledge at this moment from this graph of exactly how points in x y are clumped in time so just something to keep in mind if you're asked to talk about the time structure of the motion how many of these points occur in the first second how many of these points occur in the next second you really can't answer that question from this graph but continuing on thinking about curves representing motion in space let us pick one point on the curve at some time which we call t1 and let's draw the vector that represents marking off that point remember in the previous part of this lecture we drew a dot we gave its x y coordinates and then we represented that dot as a vector starting at the origin with its arrowhead placed where the dot had been I'm skipping the placing of the dot and simply now representing this specific point on the black curve using the arrowhead of this blue arrow now I'm going to pick another point at a different time t2 and again remember we don't know how much later t2 is than t1 we just know because of what I said earlier that this point over here on the right side of the curve occurs later in time than this point up here near the apex of the curve now I've similarly drawn the vector for this space point x2 y2 at time t2 so I have two space points two arrows representing the locations in x y of those points let's ask a question what would the arrow representing the displacement between the spatial points at t1 and t2 look like keeping in mind that t2 is later than t1 so maybe pause the video here sketch this out on a piece of paper try to draw the arrow that represents the displacement between these two points resume the video when you think you've got an answer that you're satisfied with by now you should be practiced enough to have drawn the arrow shown in red over here on the graph it begins its journey on the arrow that represents the location of this object at the earlier time and then we draw the arrow straight until its arrowhead reaches the arrowhead of the second point which is the later location in space of this object at the later time t2 this red arrow is the displacement arrow and we could denote this as delta r vector with the subscript 1 2 to indicate that it it shows the displacement between these two points one and two let us now imagine shrinking the distance along the curve over which we calculate the displacement we have picked this section of the black curve and calculated a large average displacement between the first point and the second so what I'd like to do now is imagine and we'll do it here on the graph shrinking the distance along the black curve over which I calculate the displacement so we begin first with the displacement we just had that's still shown here on the left and in the next step we can begin to shrink the distance along the black curve over which the displacement is calculated so now I've kept my first location fixed but I've now considered a location in in the curve that's a little bit closer in space than the first displacement and you see here indeed the displacement arrow has shrunk and it points in a slightly different direction than the first one did I can flip between these two so that you can see that let's continue shrinking the displacement I now shrink it a little bit more by moving the point at the later time even closer in time and in space back to the first point we again see that the displacement arrow is shrinking in length and is pointing in a different direction as we continue to move closer and closer to the first point and we do this one more time and now we're getting very close the displacement is getting very small but nonetheless is still a vector that points in a direction and has a length even though it's growing shorter it still maintains its vector properties we do this one more time and now we see that the second vector is growing extremely close to the first one in fact probably on the video you'll have a slightly hard time even seeing the little arrow that I've drawn here representing the displacement between the second point and the first point but nonetheless it is still an arrow with physical extent it has length it has direction it is still a vector now as the displacement is shrinking we can think about questions for instance what other quantity is shrinking in size as we close the gap between our two vector and our one vector and what geometric quantity does the displacement vector begin to resemble more and more as the displacement shrinks okay so let's think about these two questions what other quantity is shrinking in size as I've been closing the gap between our two and our one and then what geometric quantity and think again about curved you know curved shapes and geometric objects that can be used to represent the behavior of those shapes what geometric quantity does the displacement vector begin to resemble more and more as we shrink this displacement between point two and point one so pause the video here consider these questions see if you can come up with even vague answers to some of these and we'll move on resume the video when you're ready let's answer those questions beginning with the first one that first question was what other quantity is shrinking as I close the gap between the points are two vector and our one vector and the other quantity that shrinking as we close the displacement gap is the time displacement remember what I said earlier we don't know the time structure along the black curve but we do know that points on the left side of the curve occur at earlier times than those to the right of them on the curve and so as we move point two closer to point one whatever the time structure is we are nonetheless closing the time gap between these two locations in space of whatever object is represented by this curve okay after all the black curve is made from a set of very closely spaced x y points and each of those is marked by a time t and time increases as we move along the curve to the right so mathematically what is happening is that with each closing of the gap we are sending the delta t between those two points closer and closer to zero we are probing the displacement in smaller and smaller units of time and we know how to represent that operation the shrinking of a quantity sending its value towards zero using a mathematical operator limits and specifically the mathematical limit that we're taking in this case is we're sending delta t to zero and considering the displacement vector in shorter and shorter units of time now we can put this all together with our previous approaches to displacement and time and come to an understanding of a quantity we've seen before but now in more than one dimension putting this all together what we are doing here graphically over on the left can be represented mathematically as probing the direction that the velocity vector is pointing and the way we get to that is we take delta r and we divide it by delta t and we consider that ratio in smaller and smaller and smaller chunks of time taking the limit as delta t goes to zero recall that in one dimension as we explored this we came to recognize that this is what is known in calculus as the first derivative with respect to time of a function and that function is the vector r we can denote that as dr vector dt now the time derivative of a vector yields a vector in the same way that if you were to and we'll see this maybe demonstrate it out a little bit as we do homework and exercises in class and so forth if you were to just take this object and forget for a moment that this is the first derivative with respect to time and work through it there's nothing that ever gets rid of the unit vectors that appear in the numerator of this ratio and so those unit vectors are preserved into whatever quantity results after finishing taking the limit as delta t goes to zero this all was simply the definition of velocity instantaneous velocity specifically but we shorthand that by referring to velocity so in two dimensions or in three dimensions though the limit as delta t goes to zero of the average velocity that is delta r vector over delta t is the instantaneous velocity which in calculus is the first derivative with respect to time of the vector r and that is denoted by v with the arrowhead vector hat over it we have just witnessed in this progression of looking at displacement over shorter and shorter periods of time then calculating the average velocity in those shorter and shorter periods and considering what happens to that velocity as we grow the displacement to infinitesimal times we have just witnessed a graphical and mathematical evolution that defines the first derivative in any number of dimensions in space so we can now see how calculus really does emerge quite naturally from any consideration of motion we've inescapably continued to loop back and hit this concept of a derivative that is the change in a quantity in a very small infinitesimal unit of another quantity in this case the change in in distance between two points in space as one considers shorter and shorter times over which those displacements are measured now we can do this for complex motion and we have a full vector definition of velocity and more than one dimension but what does the displacement vector for smaller and smaller slices of delta t actually represent and this comes back to the second question what geometric feature does the displacement vector in smaller slices of time actually represent graphically if not mathematically we can see the answer to this better by zooming in on the top of this black curve that we've been looking at I have simply magnified the Cartesian coordinate system to focus on the location near the apex of the curve I was showing you a moment ago the curve has exactly the same shape we've merely zoomed in and let me put some of our displacement vector stuff back on this so we've now magnified the curve and we can look more closely at these small time slices to see what this displacement vector has been doing as we shrink the displacements that we consider by shrinking the time slices we consider so here is a vector indicating the location of the first point the one at or near the apex of the curve here is a vector representing a displacement that I was showing you a moment ago this one was extremely close at the end of our progression of shrinking times on the previous slides but now that we've zoomed in we see that they're a little bit more far apart on this scale from one another and so it'll be easier to see the displacement arrow when I draw it here's the displacement arrow which I've labeled here delta r12 vector and now it's much easier to see what this displacement arrow is doing pointing from the first vector to the second vector now don't be fooled by where the arrow heads actually stick out it's sort of a byproduct of the drawing process here that the arrow heads wind up poking out a little bit further than the actual point at the end of the arrow but don't let the details distract you from the point which is that you can now clearly see the displacement between these two locations on the motion curve much more clearly than you were able to see them before I can now continue to move the second vector closer to the first decreasing again the time units over which we're considering the displacement here's delta r12 vector and again the red arrow indicates its direction and it's getting harder to see again even though I only moved this about halfway closer than where it was before that arrow is getting small again but nonetheless that displacement arrow still has physical extent it has direction it has length so these displacements are getting really hard to see even when I zoom in and so I'm going to play a little mathematical trick here I'm going to extend the displacement vector by scaling it by a number so originally the displacement vector pointed from where it's supposed to from where it starts to where it ends the first vector to the second vector arrow head to arrow head closing the gap between them but I can multiply the length of this vector by a number four now that changes its length yes can't argue with that but remember that scaling the length of a vector does not alter its direction and so I have preserved the direction of this displacement vector while sacrificing its length so that we can actually see this thing okay so now it's much easier to see where this vector starts and where it points again keep thinking about that question what geometric quantity does the displacement vector seem to represent more and more as we shrink the displacement that is shrink the unit of time over which we're considering displacements between points on the motion curve I can shrink the gap one more time and now I've had to extend the displacement vector by a factor of eight just so you can see it at all on this graph we're shrinking the displacement more and more what do you notice about the geometric relationship between the motion curve the black line and the direction of the displacement vector as we shrink the displacement but as we keep shrinking the gap between the second point and the first point and calculating the displacement in that tinier and tinier unit of time and here I've gone even close so close that you can barely see the difference between the two arrows now representing the two locations on the motion curve nonetheless it is still a finite non-zero displacement between them but I've had to extend the length of the displacement arrow by a thousand just so you can see it and you'll notice that this displacement arrow is resembling more and more as we keep going a line that is tangent to the motion curve that is a straight line that touches the tangent curve at the point represented by r1 vector the first object location in time that line only touches the curve at that point okay in fact in the exact limit that delta t goes to zero the displacement vector and thus the velocity which points in the same direction as the displacement vector is exactly tangent to the motion curve it only touches the motion curve at one place and that place is the point where we're trying to calculate the the derivative the limit of the displacement divided by the delta t in the limit that delta t goes to zero so what have we learned from considering motion in two dimensions and very small displacements we've learned that displacement is very clearly itself a vector which we've written as delta r vector and it has both direction and magnitude and this displacement in the limit of very small units of time between the space points becomes the tangent to the motion curve at the initial point of motion we can then consider very small displacements in the limit that the time over which we're measuring these displacements goes to zero and if we take that displacement and divide it by the very small time over which we're making this consideration we arrive at the general that is two or three-dimensional definition of velocity the limit as delta t goes to zero of the difference between the r vectors at a slightly later time t plus delta t and the original time t as we send delta t to zero this looks exactly like the definition of the first derivative we explored a few lectures ago and indeed this is exactly defined as the first derivative with respect to time of the vector r so if i were to give you the functional form of r as it varies with time that is how the blue arrow varies in direction and length with time as we move along the motion curve you in principle can employ this equation and solve for the answer what is the velocity at any time t instantaneous velocity very powerful tool so we've come to understand that the first derivative of displacement with respect to time not only defines velocity and points in the same direction as the tangent vector to the motion of the curve we have we have seen how powerful calculus is and why it is so utterly necessary it is inevitable when you study motion especially on small time scales that is any small unit of displacement we are stuck with calculus and that's a good thing because calculus is an extremely powerful tool for coming to a better understanding of the natural world and without it we would not have modern engineering we would not have modern data science we would not have modern computer science all of these things rely on basic ideas about small changes in one quantity when another quantity changes by a little bit it's a very simple but very powerful notion that gives us tremendous information about the natural world we can put this all together and i'm not going to go into all the details of this because it would just be a repetition of everything i just showed you for displacement time and how you get to velocity from those concepts but we can combine displacement and velocity and time and get acceleration we have all of these quantities now in many dimensions two and three spatial dimensions are the ones we care about so for all from the efforts that we put into this so far we know that a point in two dimensions can be expressed as x comma y a vector starting from the origin and pointing to that location in space can be written as follows the vector r is given by x i hat plus y j hat in two dimensions where x and y are the coordinates marking the location of the arrowhead this vector is implicitly beginning from the origin zero comma zero the displacement between any two such vectors in space is also a vector and it results from subtracting two position vectors so if we have a position at some later time t2 we call that r2 vector and we have the position at the earlier time t1 we call that r1 vector the displacement delta r vector is given by r2 vector minus r1 vector now if we're considering extremely small units of time delta t approaching zero then we can rewrite this if for instance we we can know the functional form how r depends on time as time changes we can write this as the function r at some time t plus delta t minus the function at the earlier time t and then if we consider small displacements in time we can arrive at the definition of velocity velocity is merely the first derivative in time of the displacement vector and it points in the direction of displacement so wherever displacement points velocity points this is extremely useful if you know the displacement you may not yet know the magnitude of the velocity but you can certainly tell me its direction and of course acceleration is just a riff on this acceleration would then be the first time derivative of changes in velocity so velocity displacements if you will that's also a vector the change in velocity points in the direction of acceleration so just as velocity points in the same direction as displacement in space acceleration points in the same direction as changes in velocity or displacements in velocity space if you want to think about it that way and again acceleration or instantaneous acceleration is merely the limit as delta t goes to zero of v treated as a function of time evaluated at t plus delta t where t is the initial time and delta t is the small amount of time we're going to add to that to look at that velocity a little bit later minus the velocity at the original time t divided by delta t this is the definition of the first derivative of velocity with respect to time in fact the definition of the first derivative of something with respect to anything else is something in the numerator with this structure dependent on the anything else divided by the anything else and the limit that it goes to zero this is a very generic formula for first derivative you can determine the form of the first derivative of anything with respect to anything else by using this formula so let's take a look at example of applying calculus to a specific functional form for the vector r where r is a function of time so consider a position that changes with time in the following way we have x as a function of time the x coordinate is given by j times t t is time the y coordinate at any time t is also a function of time and it's given by k times t but squared so t is squared in the y equation of motion now what are j and k they are merely constants of the motion they are numbers whose values do not depend on time whatever j is at time zero it's the same value at time a thousand seconds 10 seconds 15 seconds 55 seconds it's a constant its value never changes now of course the units of j and k are not the same because we have to get position on the left hand side t has units of seconds what would the units of j be because we have to get position on the left hand side for y but now t is squared so we have units of second squared here what units of k would you need in order to get meters for instance over here on the left so you think about that and you'll you should arrive at the conclusion that while j and k may be constants they don't have the same units and that's okay they're there to balance the equation and make sure we go from the correct units on the right side to the same units on the left side now to answer the question what is the velocity vector at any time t we need to consider very small nearly infinitesimal changes in time relative to the original time t for which we want to answer this so we want to know the velocity at time t but to answer that question we need to consider what happens to position at t plus delta t where delta t is some tiny little add-on to the original time t and compare that to what was going on at the original time t and then send delta t to zero so we're going to write the vector r vector as a function of time we're going to write the velocity as a exercise in computing the limit as delta t goes to zero of the displacement divided by delta t and then we're going to consider the changes in such a limit so let's begin the vector r would be given by the x component times the unit vector i hat plus the y component times the unit vector j hat well we know what x of t and y of t these functions of time x of t is jt y of t is kt squared and so we just make the substitution and we arrive at the final form of the vector r as a function of time it is jt i hat plus kt squared j hat that's as far as we can go with this there is no further simplification we can make to make this look any prettier or any less complex so let's move on to calculating the displacement delta r as a function of time for two closely spaced times t plus delta t and just t well that's going to look like this the displacement is going to be the difference in the x positions times i hat and the difference in the y positions times j hat and all of that will be added together well we need to evaluate x at the time t plus delta t and we need to evaluate x at the time t and similarly we need to evaluate y at the time t plus delta t and at the time t we do that in the next step we are going to have x of t plus delta t is going to be j times the time quantity which is t plus delta t and then x of t is just j times t then we have y of t plus delta t well that is going to be k times the time quantity squared and that is t plus delta t squared and we're going to subtract from that y at the time t which is just k t squared now we have the unit vectors still out in front of all this we have i hat over here and j hat over here and all of this has been added together still so all we've done is substituted for the values of the functions at the respective time quantities t plus delta t and t next we want to write down the definition of the velocity our goal is to find the velocity at any time t so to get the answer to that question we need to take the limit as delta t goes to zero of the displacement vector divided by delta t well i'm not going to work out all the algebraic steps for you i would recommend that perhaps you pause the video at this point and make sure that the following equation is a sensible result of what happens next but the bottom line is that we wind up after substituting in with the vector delta r which we calculated here and dividing by delta t we are going to wind up expanding out a polynomial in the numerator cancelling out a bunch of terms that look like each other but have opposite signs in front of them so one is positive one is negative and then we're going to be left with this quantity j delta t in the i hat direction plus k times this quantity 2t delta t plus delta t squared in the j direction all divided by delta t and so this delta t in the denominator is going to cancel this delta t here and one of the delta t's that can be pulled out of this full polynomial and in the end when we send delta t to zero we're going to find out that the only terms that survive the limit are an x term j i hat and a piece of the y term 2kt j hat this is the equation for the velocity at any time t what do you notice about the x component of the velocity and the y component of the velocity well one observation is the x component does not depend on time the velocity in the x direction is constant in time at any time t the velocity along x is the same it's j this is different than the y term in the y direction we see that the velocity has an explicit dependence on time time appears in here multiplied by the numbers 2 and k so in fact as time evolves the velocity component in the y direction will change but it has no effect on the x component the x component of velocity and the y component of velocity don't talk to each other so the x component continues along at a constant rate while the y component varies with time in fact it increases with time as time gets bigger the y component gets larger this would suggest there's something accelerating in the y direction the motion of this object so let's review the key ideas that we have explored in this lecture we have combined one-dimensional descriptions of motion position and time displacement velocity acceleration with the vector concept we developed in vectors part one and we've extended our description of motion to three dimensions the concepts of average and instantaneous quantities have also been extended into three dimensions now we've used two dimensions as an example but nothing i've done on the previous batch of slides couldn't be repeated with a three-dimensional example by adding a term with a k hat unit vector in it and a number multiplying that representing the length of the of the component in the z direction everything can be repeated again for that and you'll see that in in terms of format nothing changes the problems may get more complex because you have one more dimension to deal with but the format in which you solve those problems in which you describe those motions is identical such motions in many dimensions is certainly not simple and we're going to begin to explore kinds of two-dimensional motion which at first may seem quite complex but through the lens of the mathematics that we have come to develop here we will begin to see that it's not as bad as it seems if Sumi you can make simplifying assumptions about what forces are acting in certain situations and you'll begin to pick apart even seemingly complex motions and even under the situation of approximations get very useful answers out of those mathematical approaches to understanding the motion we'll get that because we have a toolkit of space and time calculus and vectors that we can bring to bear even on motions that seem as complex as comets swinging around the sun and planets orbiting in regular and nearly circular patterns as they revolve around the parent star