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# Introduction to one-dimensional motion with calculus

AP.CALC:
CHA‑3 (EU)
,
CHA‑3.B (LO)
,
CHA‑3.B.1 (EK)

## Video transcript

what we're going to do in this video is start to think about how we describe position in one dimension as a function of time so we could say our position and we're going to think about position on the x axis as a function of time and we could define it by some expression let's say in this situation it is going to be our time to the third power minus three times our x squared plus five and this is going to apply for our time being non-negative because the idea of negative time at least for now is a bit strange so let's think about what this right over here is describing and to help us do that we could set up a little bit of a table to understand that depending on what time we are let's say that time is in seconds what is going to be our position along our x axis so at time equals zero X of 0 is just going to be 5 at time 1 you're gonna have 1 minus 3 plus 5 so that is going to be C 1 minus 3 is negative 2 plus 5 is going to be we're going to be at position 3 and then at time 2 we are going to be at 8 minus 12 plus 5 so we're going to be at position 1 and then at time T equals 3 its gonna be 27 minus 27 plus 5 we're going to be back at 5 and so this can at least help us understand what's going on for the first 3 seconds so let me draw our positive x axis so say it looks something like that and this is x equals 0 this is our x axis x equals 1 2 3 4 and 5 and now let's let's play out how this particle that's being described is moving along the x axis so we're gonna start here and we're gonna go 1 2 3 let's do it again we're going to go 1 2 3 the way I just moved my mouse if we assume that I got the time roughly right is how that particle would move and we can graph this as well so for example it could it would look like this we are starting at time T equals zero our position are what this is our vertical axis or Y axis but we're just saying Y is going to be equal to our position along the x axis so that's a little bit counterintuitive because we're talking about our position our position in the left/right dimension and here you're seeing it start off in the vertical dimension but you see the same thing at time T equals one our position has gone down to three then it goes down further at time equals two our position is down to one and then we switch direction and then over the next if we say that time is in seconds over the next second we get back to five now an interesting thing to think about in the context of calculus is well what is our velocity at any point in time and velocity as you might remember is the derivative of position so let me write that down so we're gonna be thinking about velocity as a function of time and you could view velocity as the first derivative of position with respect to time which is going to be equal to we're going to apply the power rule and some derivative properties multiple times if this is onion if this is unfamiliar to you I encourage you to review it but this is going to be three T squared minus 6t and then plus zero and we're gonna restrict going to restricted domain as well for T is greater than or equal to zero and then we can plot that it would look like that now let's see if this curve makes intuitive sense we mentioned that one second two seconds three seconds so we are starting moving to the left and the convention is if you're moving to the left you have negative velocity and if you're moving to the right you have positive velocity and you can see here our velocity immediately gets more and more negative until we get to one second and then it stays negative it's getting less and less negative until we get to two seconds and at two seconds our velocity becomes positive and that makes sense because at two seconds was when our velocity switched your directions to the rightward direction so our velocity is getting more and more negative less and less negative and then we switch directions and we go just like that and we see it right over here now one thing to keep in mind when we're thinking about Val city is a function of time is that velocity and speed are two different things speed speed let me write it over here speed is equal to if you think about it in one dimension you could think about it as the absolute value of your velocity as a function of time or your magnitude of velocity as a function of time so in the beginning even though your velocity is becoming more and more negative your speed is actually increasing your speed is increasing to the left and your speed is decreasing you slow down and then your speed is increasing as we go to the right and we'll do some worked examples that work through that a little bit more now the last concept we will touch on in this video is the idea of acceleration an acceleration you could view as the rate of change of velocity with respect to time so acceleration as a function of time is just going to be the first derivative of velocity with respect to time which is equal to the second derivative of position with respect to time it's just going to be the derivative of this expression so once again using the power rule here that's going to be 60 and then using the power rule here minus 6 and once again we will restrict the domain and we can graph that as well and we would see right over here this is y is equal to acceleration as a function of time and you can see at time equals zero our acceleration is quite negative it is negative six and then it becomes less and less and less negative and then our acceleration actually becomes positive at T equals one now does that make sense well we're going one two three you might to wait we didn't switch directions until we get to the second second but remember after we get to the first second are our velocities our velocity in the negative Direction becomes less negative which means that our acceleration is positive if that's a little confusing pause the video and really think through that so our acceleration is negative then positive and then positive continues and so this is just to give you an intuition in the next few videos we'll do several worked examples that help us dive deeper into this idea of studying motion and position into this idea of studying motion in one dimension