On this page of notes, I'm going to show the derivation for two equations. They apply to constant acceleration, so they are appropriately named the constant acceleration equations. And I am going to do the derivations graphically for you. You do not need to memorize this as a proof, either for our tests or for the AP tests, but I believe it's useful and... comforting to know where an equation that we are going to use over and over and over again all year long where that equation actually comes from.
So let's do it. We have constant acceleration, so that's where we're starting. We're starting out with a constant acceleration.
So we have acceleration as a function of time and look the acceleration is constant so it remains constant. So If I want to go from an acceleration graph to a velocity graph, I'm going to utilize an area calculation. So we get that delta v, right?
Delta v, right, is going to be base times height. So this we've just talked about in our notes. So we get delta v equals base times height. Now, My delta t, I am going to generalize my t initial to be 0. I can always start my stopwatch at time t equals 0, and I'm going to drop the subscript off of t final and just make it a t. So delta v, v final minus v initial.
If I drop the subscript off of t final and use 0 for t initial, divide it by t, I cross multiply, I bring my t here up with my a, and I get my constant acceleration equation. There it is. I rearrange it a little bit to get v final equals v initial plus acceleration times time.
And this is really important. So make sure you get this into your notes for the constant acceleration equation. v final equals v initial plus acceleration times time. You must, and must means must, must use positive or negative for both the v's and the a. V final, V initial, and A.
Of course, time is always positive. Now, when we plot the velocity graph, we don't know the initial position. We must be given it.
So I left V initial as a general unknown variable, V initial. Now, for V final, notice what we've done here. For V final, we've replaced it with a generic V as a function of T.
So V as a function of T will represent whatever. final velocity value we are at. And here I just have it generally shown as a v final. So we get our v as a function of t.
It's a linear function. And when we want to go from a velocity graph to a position graph, we again are going to look at area. And now to calculate my area, I broke it up into a rectangle, right, and a triangle.
So we have the total area. below the V of t graph, so my V of t graph is here. The total area between the V of t graph and my horizontal t equals zero line, remember that's we always calculate the areas to the horizontal zero line, is delta xA plus delta xB delta xA is a triangle, one half base t times height v final minus v initial, so notice the height is not v final, but it's v final minus v initial, plus the area of delta xb, which is the area base times height v initial.
When we do our algebra, we get our second constant acceleration equation. Delta x equals 1 half a t squared plus v initial. t squared is a quadratic, so our...
Position function is actually quadratic. We'll be talking more about that for constant acceleration problems. But we again have our second equation that we will be using constantly through this entire course.
Very important equations for us to have memorized. We don't want to look them up off of lookup tables or rely on anything other than our own knowledge of these equations to write them down very quickly. We have delta x is x final minus x initial.
I brought the 1 half at squared, so I rearranged the equation a little bit to put the equation into its conventional form. Where we have the boxed equation here is the conventional form, x final equals x initial plus v initial times time plus 1 half at squared. And as it says, that I circled here, I'm trying to really emphasize the importance of, because you will not calculate the right numerical answers if your x's, your v's, and your a's do not have the appropriate forward positive, backward negative, or for position, positive side of the x-axis, negative negative side of the x-axis for those x variables. So there you have it.
We are going to use those two equations, right? The final equals the initial plus acceleration times time. And the x final equals x initial plus v initial times time plus 1 half a t squared throughout the entire year. So while you don't need to know the derivations, have them memorized, it is nice to see that they actually come from a logic flow and reasoning that we have been using all along.