in this video we return to the motivating example that we started our vector-valued functions unit on and that was the idea of tracing a three-dimensional particle through space here a spaceship where the trajectory is defined by cos of t sine of t and e to the minus t we now recognize that as a position vector over time and we know what we can do with that in terms of representing it sketching it and then analyzing properties like the velocity and acceleration first off let's start with an analysis of the trajectory itself because it's in 3d it's a little tougher to visualize but what we can do very easily is decompose this into the x and y components and there we see our favorite circle the unit circle unit speed so what we are going to see if we look down from above a top view of the trajectory is a particle moving around in a circle over time in addition to that though we also have a z component and that z component is following an exponential decay e to the negative t now in the previous page we indicated that we would be starting at time zero this time so no negative time values allowed but we're going to have an exponential decay for the z value starting at time zero which is z equals one so z at time zero is equal to 1. let's draw this in a similar way to the trajectories limiting ourselves just to the z values so call these are z axis and that's 0. here we're going to have z equals 1 and then it's going to be a trajectory that goes down and down and down we only have one dimension here so it's just going to be tracing out an arc where it's decreasing over time but it never quite reaches zero so our z values are going to start at one and then go down quickly and then slow and slower is lower and slower and slower if we combine that with let's just say x here let's make this a little more interesting if you imagine your x starts at 1 we would have something like this a helix we'd be going down and down and down and down and down and if we look at a side view of that expression a side view of that trajectory we would see the side of a spiral helix type structure but with an inconsistent pitch a pitch that's getting tighter and tighter as we get closer and closer to the axis so top view just looks like a circle side view we'd see that side of the circle and then the z component decreasing and decreasing as we get closer and closer to zero but never quite reaching the plane so that's a fairly intricate construction of the trajectory given a relatively simple formula so this is a very general and broad tool for defining 3d trajectories this vector valued function idea of course once we have the formulas we have some visualization of the position over time we can ask questions like what's the speed of this particular spaceship at a particular moment so of course speed requires velocity which is the vector quantity and we know how to get that now we simply take each component and differentiate it derivative of sine is cosine and the derivative of the exponential is e to the minus t and then another negative sign as the chain rule output speed is the magnitude of the velocity vector and so that's the square root of each component squared added up together so negative sine of t all squared plus cos of t all squared plus negative e to the negative t all squared if we take a careful look at that the negatives inside the square negatives inside the square what we get is square root of sine t all squared sine squared t plus cos squared t and then here we end up with e to the negative t squared is the same as e to the negative two t and we actually don't need brackets around that that's reasonably clear all square rooted well we can go one step further we don't have to this is the speed but we also notice that sine squared plus cos squared equals one so our speed as a function of time is given by that expression there at time t equals 0 we get a speed of square root of 1 plus e to the minus 2 times 0 e to the 0 square root of 1 plus 1 square root of 2 meters per second so we now have the ability to go from a position vector valued function or parameterized space curve to velocities and speeds we can also look further and go down to accelerations notice here that we'd be talking about the magnitude of the acceleration and for whatever reason the idea of velocity being a vector and speed being its scalar or magnitude quantity never really translated into accelerations accelerations we kind of use interchangeably as the magnitude or the vector in this context it makes sense to look at it as the vector quantity first and then we'll talk about the magnitude as a separate operation recall from the last slide that the velocity we had was negative sine t and cos t and negative e to the negative t for the z component and so our acceleration is going to be the derivative of that we're going to take a time derivative and we're going to get ourselves a negative cos t a negative sine t and a double negative e to the negative t or back to positive now this question asked about the greatest acceleration so greatest magnitude so we need the magnitude of the acceleration and if we look at these ingredients they're actually all the same as the last one here we had sine cos and an exponential and the plus and minuses all went away because of the squared so we'll write it out formally once just to see what the building blocks are but we will jump to the simplified form a little more efficiently this time because again we're going to have co squared plus sine squared and then the exponential the co co-squared and sine squared are going to give us a one and then the e to the negative two t squared or e to the negative t squared gives us e to the negative two t there we go so the magnitude of the acceleration is given by this formula here now you might be tempted to do some kind of optimization strategy that kind of thing that is certainly effective but here we actually have enough information to answer the question now we have t only greater than or equal to zero in our consideration and we know that this decreases with t it's an exponential decay because of the negative sign there so as t increases then this magnitude of acceleration is going to be one plus something keeps getting smaller and smaller and smaller and so that means we should pick the smallest possible t to maximize our acceleration and for our particular scenario here where the smallest t that we can pick is zero we're going to get the largest acceleration at time zero and being very specific here the magnitude is going to be largest in that domain so this is conditional on that time values the time values we're allowed to use being only greater than or equal to zero and if it's helpful you can tie that back to the diagram the idea of being spun around the spinning is always a constant acceleration of magnitude one whereas this particle is initially moving down fairly quickly but then the acceleration is upwards you can check the sign in the calculations and it's slowing down the most quickly that has the most upwards acceleration at the first instant that we're considering once it gets to a flat trajectory down here it's not actually changing its velocity very much there's very low acceleration so again it makes sense from the graphical perspective that the first time point we have is when the acceleration magnitude is going to be largest let's take that one step further imagining that spiral shape again only this time we're going to freeze frame a little later at time one and we're going to follow a trajectory after that that will be a straight line so we saw an example of this earlier in the notes that line we're going to call l of t and remember how we work that in the earlier examples it's going to be our position plus our velocity and let's call this position 1 and times the velocity at time one times time so if we work that through how do we get our position we have a formula for the position how do we get the velocity we have a formula for velocity as well we just sub in the values so if we submit our position at time one that'll be cos of one sine of one and e to the negative one that's our position at the falling off time and then we add to that our velocity vector which we had a couple pages ago here we go negative sine cos negative e to the negative t so negative sine of 1 cos of one and negative e to the negative one and we multiply that by time and so that defines our trajectory now there's a little bit of an inconsistency here it's worth pointing out that we use time equals zero kind of twice here we have time equals zero and time equals one on the trajectory on the curve and then we essentially redefine this point here as time zero on the l of t curve so in that case here we had time zero on l of t so just be aware that you might need to re the time values to define this trajectory if you want it in the same zero time as the other part for that level of detail is something we're not going to be too concerned with in this class as a final jumping off point for this imagine this question here if you follow that straight line trajectory you've got the curve you launch at time equals one and you follow a straight line afterwards this is all in the context of a three-dimensional universe imagine tracing this line out maybe with a bit more angle on it let's pick something that looks visually a bit more interesting if you follow that line eventually it's going to intersect the x y plane that would be the landing point for you if you after you jump off the spaceship the question is what is that location on the xy plane it's something you can definitely work out using what we have in the previous page but it's worth exploring a bit on your own