funding for an audible paint course television was made possible in part by the College Board almost nobody can reason with you Mr Wares Cunningham McHugh the great founder itself and all other funding were made possible by viewers like you we forced you [Music] oh hello and welcome back to another episode of the APC TV the mechanics Edition I'm your old sister McHugh and today you'll be watching season one kinematics episode 3 One Direction not the band feat calculus so just for today's videos to be able to explain how to calculate position velocity and acceleration using calculus meaning both derivatives and integrals and describe where your three Canon cues come from and no unfortunately I could not get One Direction the actual band of combo we have our friend calculus accompanying us on this journey in this episode so in physics you need to know how to do derivatives and integrals and we will go along that journey together but we need to know what a derivative is and derivative is really really the rate of change of something so almost like if you're going to graph it's the slope so a derivative goes with rate of change and slope almost like a word association game and we have definitions that use the words rate of change like velocity and acceleration so if you're given some position function X of T maybe it's 5T squared plus 6t plus two it's weird it changes a lot with time if you take the derivative or the rate of change that gives you the velocity so if you have position you take a derivative to get velocity and we call that DX over DT or X Prime but we can go further meaning acceleration is the rate of change of velocity so if you still have a velocity function let's say 10t plus 6. if you take the derivative of that or the rate of change it gives you acceleration or DV over DT or V Prime as a function of t and the reason we have to use calculus is because things in physics don't always stay the same yes we have constant accelerated motion we have constant velocity but things do vary as a function of time and if you want to figure out how velocity and acceleration respond with time we will most likely need to take a derivative of the position function to get velocity and a double derivative of position to get acceleration which means you take the derivative twice or you simply just take the derivative of velocity because we need to find how it varies with time that is the reasoning behind using calculus and the most common one we see is the power rule for both derivatives and in a moment I'll talk about integrals but all the power rule is with derivatives is if you have some function to the nth power so let's say x to the second power to take the derivative you multiply by the power so if it's x squared you multiply it by two and drop the power by 1 to give you two x so if the original function was x squared you multiply out the two and do two minus 1 for the power and you're left with 2X and this is the more formal way of saying the power rule it is on the APC reference tables but is one of the more common bits of derivatives and calculus you'll see is the power rule on the flip side if you could take a derivative to go from position to Velocity to acceleration you could take an integral or an antiderivative to go back up so if you have an acceleration function or an acceleration constant if you take the integral or really what we think of as the area if you're thinking about it graphically if you take the integral you get velocity so the integral of a of t with respect to time gives you velocity but now you need position you take the integral one more time position is the integral of V of t with respect to time and that DT just means with respect to time and again if you're not comfortable with integrals think of it as an antiderivative what goes down must come up and I know the actual expression is what goes up must come down but it's the same logic if you could take a derivative to go from X to V to a you could take an integral to go from a to V to X or an antiderivative and another reason it's called an anti-derivative is because you should end up back at your original function after taking the derivative and antiderivative an example the power rule again the most common form of calculus for us is if you take the integral of some function to the nth power your integral is X to the N plus 1 over n plus one so let's go back to that previous example here I'm blocking it of the derivative we took the derivative of x squared and got 2X well if it's the antiderivative the integral let's prove it brings us back up to x squared so the antiderivative or integral of 2 x well you add 1 to the power so x to the first Power Plus 1 is x squared divided by n plus one your new power is two so 2x squared over 2 brings you back to x squared so if you're not comfortable with the concept of an integral think of it as an antiderivative you could take the derivative of a function and the antiderivative should bring you back and you're really raising the power and then dividing by that new power and for at least power rule but this is an important um these are important relationships to know that X to V to a is a derivative and a to the E to X is integral or area and to give you a concrete example of why this matters for kinematics is the Cami cues and their origin story so if you have constant accelerated emotion you have some constant acceleration constant a okay right you start off with some constant number 5 10 20 negative 30. you have some constant acceleration so what's the velocity function if you have constant acceleration well if you take the integral of acceleration with respect to time of some constant number a you really have a t to the 0 with power which is 1 a t to the zero any power is one so add one to T to the zero and you get a t right because you do x to the N plus 1 over n Plus 1. so a t to the 0 integrated you get a t to the first over one which is a t but then you have what we call some constant of integration where if you're doing an indefinite integral you have some potentially constant initial condition but wait that initial condition is just our initial velocity so you end up with a t plus v naught or wait V equals V naught plus a t right so now you already have one of the Cami cues no I may not go off for him to explain this one the second candy Cube that comes from calculus is the position function if you now know your velocity function you can integrate it or take an antiderivative to get position and so the integral V naught which is a constant plus a t with respect to DT well V naught t to the zeroth power becomes V naught t to the first over one or V naught times t a t to the first becomes a t squared over two or plus one half a t squared plus the constant of integration but that's just going to be your initial position so wait that just gives you X of T equals x naught plus v naught t plus one half a t squared so the origins of two out of three can accused simply lie in the fact that if you have a constant acceleration you can integrate to get a velocity function V equals V naught plus a t and you can integrate a velocity function to get a position function x equals x naught plus v naught t plus 1 f a t squared and there is a third candy Cube but that Origins remains a mystery for now we will discuss the origins of the third candy Q when we get to energy there is an origin story for it there always is and we'll get to it later but two out of your three chemicals are simply from calculus so if you have any questions please feel free to email me or leave a comment otherwise have a fantastic night