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# Parametric equations intro

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

## Video transcript

let's say I have a cliff let's make this cliff I don't know miss say it's 50 meters high 50 meters and on this cliff I have a car and this car is not just sitting on the cliff it's driving off of it a very dramatic problem so let's see if this car here and it's driving off of this cliff at five meters per second five meters per second and I want to know what is the path of this car as it falls off of the cliff so let's set up a little coordinate axes here say that I get my y-axis let's say that this is the y-axis right there and then this will be my x-axis x-axis so this is y this is X let's say that this is the point well we know this is a 50 meter high cliff maybe y equals zero is sea level so this is this would be 50 right here and let's say that this point right here on the cliff that said X is equal to X is equal to 10 so this point right here is the point 10 comma 15 let's say the car is right at this point right about to drive off the cliff at time is equal to zero so this is at time equal to zero time equal to zero so T for time time is equal to zero so my question is what happens to this call car as it drives off the cliff so this is a bit of a physics problem and I won't go deep into the physics and I won't prove some of the equations and I encourage you to watch the kinematics videos the projectile motion videos if you want to know where the equations come from but the point here is just to get the equations and see what the graph looks like so if I were to know X as a function of time X is a function of time let me know suitably vibrant color so X as a function of time is going to be what well we're going to assume that we're on a on a planet that has no air where in a vacuum so if we start off in the X direction at 5 meters per second to the right we won't be decelerated by air or friction or anything else Newton's laws of motion an object in motion stays in motion unless it's a unless it's unless it's affected by a net force and there won't be any net force in the X direction so it's just going to keep moving to the right at five meters per second and position or distance it's just equal to velocity times time our velocity is five times time and of course it didn't start at tot at X is equal to zero right this is at time equals zero so it started at X is equal to ten so you want to know it's kind of X of zero where it started off so plus ten and this should make be a little intuitive for you right at time is equal to zero this term cancels out we're at X is equal to ten that that makes sense at time is equal to one we should be a little bit will be five meters further out so on and so forth fair enough that's X as a function of the parameter time as you probably realize that this is a video on parametric equations not physics so it's nice to early on say the word parameter parameter parameter and time tends to be the parameter when people talk about parametric equations although it could be anything it could be radius or or angle or who knows what else so let's figure out what Y is as a function of time so why Y as a function of time it's going to be equal to the initial Y position or Y of 0 which is 50 where 50 meters up in the air 50 meters up in the air plus the our initial velocity in the Y direction and we don't actually have any velocity in the Y direction the car isn't jumping or isn't diving it's it's just moving horizontally to the right and and the cliff is supporting it so it's not it has no Y velocity but if you were curious it would be the Y velocity times time but since there's no Y velocity times time at least initially I'll put nothing there plus the acceleration of gravity times so times x squared over 2 we want to figure out the sign and just just so you know I mean it's nice to touch on the physics a little just so you know where these formulas come from and you know the motivation behind why you would even use a parametric equation gravity goes downwards in this example and downwards in this example is in the minus y direction with y is decreasing and the the real and you know it's not exact but you know gravity is normally 9.8 meters per second squared in most textbooks but for the sake of simplifying this we'll say that it's approximately 10 meters per second squared that's how fast everything will be accelerated downward on this planet since it has no where let's assume it's a planet with a little bit more mass than Earth and since it's going downwards this direction is negative so in our formula up here it's our initial position we had no velocity times time so I won't put that there minus 10 meters per second squared times T squared over 2 and you can watch the projectile motion videos to figure out how I got these these formulas right there but that's not the point at this the point of this is to graph what happens to the cars and learn a little bit about parametric equations so what is the path of this car as it falls off the cliff let's make a table here let's make a table and I'll do the so x and y are a function of this third parameter T so we're going to set T at different values and we're going to figure out what x and y are equal to and I'm just going to arbitrarily pick some T's T is equal to 0 T is equal to 1 2 & 3 at time is equal to 0 what is X X of 0 this is 0 X is equal to 10 meters at time is equal to 1 what is X so this is X of 1 right if I wanted to write that notation so 5 times 1 is 5 plus 10 is 15 X of 2 5 times 2 plus 10 that's 20 and it makes sense every second we're getting 5 meters more to the right or X is increasing by 5 meters so a neck T is equal to 3 15 plus 10 is 25 easy enough the Y is a little bit more complicated and just to simplify this it's the same thing as 5 right 10 divided by 2 so 50 minus 5t squared so time is equal to zero this term cancels out we just have 50 meters up in the air at time is equal to 1 1 squared is 1 times 5 is 5 50 minus 5 is 40 or 45 meters in the air is that right right yeah time 150 right and then at time is equal to 2 2 squared is 4 4 times 5 is 20 50 minus 20 is 30 and then finally at time is equal to 3 9 SI finally because that's the last number we picked time is equal to 3 3 squared is 9 9 times 5 is 45 50 minus 45 is 5 so let's plot these points so time is equal to 0 that's that's what we got right there at time is equal to 1 where X is equal to 15 that's roughly you see this is 5 10 15 let me do all of them 15 20 25 and then the y-axis let me label that while we're at it so this is roughly 10 20 30 40 50 so time is equal to 0 where 10 50 that's that point right there at time is equal to 1 where it 15 45 so X is 15 Y is 45 which is right about there so this is T is equal to 1 at time is equal to 2 or at the at the coordinate 20 30 so 20 and 30 it's right about there so this is at time is equal to 2 and then at time is equal to 3 we're 25 5 so we're right there 25 5 time is equal to 3 and if we kept going on at some point we're going to hit the ground you can figure out actually set this equal to 0 and you figure out the exact time you hit the ground actually let's do that if if this is equal to 0 50 you get T is equal to the square root of 10 which is a little over 3 seconds which makes sense right a little over 3 seconds we're going to be hitting the ground but anyway what's the path of this car what's going to look something like this it's going to look like this it starts getting accelerated downwards and then plunk it hits the ground at three-point-something seconds now what was interesting here is that by setting the parameter not only did we get the curve right we got this curve which is kind of half of a parabola half of a downward shaping parabola and we can actually eliminate the T and just get the equation for that parabola and we'll do that in future videos but what was interesting by making it a parametric equation we know the direction of the car if you just saw this graph without the car and everything else I drew you wouldn't know which way the car is falling but now we know that as T is increasing we're going in that direction so we can draw some arrows here so because it's a parametric equation we can draw some arrows and then the most important thing is we know exactly where the car is at any time T you can substitute T is equal to 1.25 seconds and you know exactly where the car is so you can plot these points and you can kind of get a sense that as as time goes on we're getting accelerated downwards and that's why for every second further the especially the Y distance gets further and further apart anyway I just wanted to give you this example this wasn't although this was a good physics problem the intention wasn't to teach you physics the intention is to give you the motivation behind why parametric equations even exist parametric equations this these two things are parametric equations we defined x and y as a function of a third parameter T instead of defining Y in terms of X or X in terms of Y like we've done every other time since then and this is super useful I mean you could imagine when you have really hard physics problems where you want to figure out the three-dimensional position of something then you'll have X as a function of T Y as a function of T Z is a function of T all sorts of interesting problems come out of using parametric equations not just in physics but anyway I thought a good place to start is the motivation because the first time I learn parametric equations like why mess up my nice and simple world of X's and Y's why introducing a third parameter T this is why because you can figure out the path of things you can figure out the direction of something as it moves along a curve and you can figure out the exact position at any in this case time
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