equations of motion

Before we look at equations, a quick note about notation: any time you see a subscript "zero," it indicates an initial condition. This zero is often called "naught." In class, if you hear me say "v-naught," I'm referring to v0: the initial velocity of the object. That is, the velocity at time t = 0 seconds. As I mentioned on the home page of this lesson, you can calculate the position and speed of an object at any moment in time given only its initial conditions!

Also, don't freak out when you see a TON of subscript x's. Traditionally, "x" is reserved to represent the horizontal direction and "y" to represent the vertical. These subscripts are only there to remind you that these equations are valid only in one direction at a time. (A vertical acceleration does not affect a horizontal velocity, and so forth). When we get to two-dimensional motion, we'll see these same equations with little subscript y's instead. If I'm working a problem with only one direction to worry about, I'll often omit the x and y subscripts entirely, but I thought I'd include them here so the equations I walk through with you match those on the AP equation sheet.

SUPER IMPORTANT!!! The following equations assume a constant acceleration. If you were in my physics class last year, these were the equations of motion we always jumped to. Now that you're a level up, we can only use these equations if the object has a constant acceleration. If the acceleration changes with time, these equations are NOT valid.

Position Equation

x is the position at any given moment. Sometimes it's referred to as the "final position" (Units = meters)
x0 is the initial position - also known as your starting position from some reference point (Units = meters)
vx0 is the initial velocity (in the x-direction) - how fast the object was moving at the start, when t = 0 seconds (Units = meters/second)
t is the amount of time that has passed. It starts at zero and then progresses normally (Units = seconds)
ax is the acceleration of the object (Units = meters/second^2)

Velocity Equation

vx is the velocity at any given moment. Sometimes it's referred to as the "final velocity" (Units = meters/second)
vx0 is the initial velocity (Units = meters/second)
ax is the acceleration (Units = meters/second^2)
t is the amount of time that has passed (Units = seconds)

Combo Equation

*Disclaimer* - I'm pretty sure I made up the name "Combo Equation." You likely won't find it if you search for it under this name, but I thought it was appropriate. I basically solved both the position and velocity equations for "t" and set them equal to each other. This equation is useful if the amount of time that has passed is irrelevant to your given information and what you're trying to solve.

vx is the velocity at any given moment in time. Sometimes referred to as the "final velocity" (Units = meters/second)

vx0 is the initial velocity (Units = meters/second)

ax is the acceleration (Units = meters/second^2)

(x - x0) is the change in position. (Units = meters)

Just for funsies, let's look at this page's information combined with the last page's! If we assume "a" is constant, we can treat it as, well, any other constant value.

Start with the position equation. Take the derivative of it with respect to time, and voila! You have the velocity equation!

That was so much fun, let's do it again - take the derivative of the velocity equation with respect to time, and holy-corn-bread-on-a-platter, you've found that it's simply "a," meaning acceleration = acceleration. Lovely how these things work out!

This page, while containing incredibly important and useful info, is pretty boring due to a lack of images. To liven it up, here's an adorable picture I found of a deer and a cat:

Image:
Pinterest. (n.d.). Cute Overload [Photograph]. Retrieved from https://www.pinterest.co.uk/pin/488640628299733191/

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