So first off I would have had this out like I intended on Friday, however my mom had a somewhat small complication from her surgery that day and had to be kept overnight. Consequently I didn't have time.
Now then, given the (some what surprising results) from the poll I have decided thus: I actually am going to break this into a 2 diary a week series. This is probably going to be a little time consuming on my end but enough people wanted more math that I think I owe them that. At the same time I do not want to chase people away.
So the tentative schedule will be that every Friday I will put up a diary on the week's physics topic and will do so with the least amount of math I can do. A lot of it will be me presenting equations without showing where they come from. I will then in a rather hand waving manner do my best to explain the equations, concepts and so on and any interesting implications.
On Monday or Tuesday depending on my schedule I will revisit Friday's topic but with a much more rigorous mathematical outlook. I plan to also introduce and even try to explain the basic math concepts involved. But in the end I will still probably have to assume some previous mathematical background.
On the mathematical days I will do my best to hang around and provide additional explanation as needed.
Now then let's talk about equations of motion.
Equations of motions, is a fairly wide and diverse topic that can include linear motion to euler-lagrange equations.
For now we will be dealing with linear motion and motion in two dimensions. Technically speaking the equations that I am going to introduce are based off of Newton's 2nd law; that said the equations can be derived without invoking Newton's 2nd so I felt these equations were as good as any place to start.
Without further adieu here are the equations:
Now that I have assaulted people with these strange symbols and letters, let's provide some meaning to what people are seeing.
delta s = the distance between initial and final positions, this is
know as the displacement because of how DK formats things
this looks like a triangle then an s in the above equation
this is given by s-s0
V0 = the initial velocity (speed in a given direction)
v = the final velocity
a = the constant acceleration
t = the time taken to move from the initial state to the final state
s0= initial position
s=final position
Mostly I think this is fairly intuitive and so I am not going to go into much detail here. I could be wrong and if so please ask below and I will explain.
Some important things here V0 and s0 is written that way because the formatting of DK is not too impressive for dealing with subscripts. It should read v subscript zero and s subscript zero.
It is important to note that these equations are valid only for constant acceleration. It is possible to deal with nonconstant acceleration but to do so enters the realm of integration and differentiation. Thus I will not deal with that case here, instead look for me to touch upon that Monday or Tuesday.
Now then in regards to V0 and a we have to remember we are dealing with vectors. And thus we have to account for direction. In linear motion (essentially 1 dimension) doing so is easy. By convention movement to the right is positive and movement to the left is negative. So if I talk about +5 meters per second then I mean movement going in the direction of -> at 5 meters per second.
This actually bring to a 3rd important point, I know that the US by and large refuses to update to the metric system. Science or at least physics pretty much has converted to the metric system as we have to be able to share results outside of the US. There are some exceptions to this but not many. Therefore I will talk mostly in metric, there are ways to convert to english units but I won't be using them.
Now why are these equations useful? Because they allow for the solution of any problem in linear physics.
For example let's take a classic example, you walk into your backyard and hurl a baseball into the air at a known velocity. Now what is the maximum height that baseball will travel?
The answer is deceptively simple.
First let's go what we know, we know V0, we can define s0 as 0 (a note, this is a common physics trick and allows one to effectively edit out what would be a problematic variable), further we can (and should) define our linear system in terms of height. So +s means the ball is going up and -s means the ball is going down (if it helps you could of course use the variable y)
Now how do we define acceleration? That too is easy, acceleration on the earth is a near constant 9.8 meters per second squared (again there is some hand waving here, the earth's gravitation field is not strictly uniform and variation does exist however accounting for such is greatly outside of the scope of this diary). And because we have already defined our linear system, it also means that the acceleration is negative. That means we have our acceleration and that it is defined in a way that intuitively makes sense.
So we the variable we know are V0, s0 and a. We need s. Now if I was being nice I would have also have stated either the total time of the ball in the air or the time it takes to reach the apex of the throw. Why the total time you ask? Well it's hand waving but simply put, the total time the ball spends going up must equal the time going down. Thus half of the total time is equal to the the time needed to reach the apex of the throw.
With that information it would be very simple to solve the problem as we could just plug into the last equation.
However why make things easy? And besides this allows me to highlight a contention of mine about physics. Namely that while one certainly needs a lot of math to understand the mathematical proofs behind the equations physics uses, the fact is that a lot of physics is about being creative and thinking outside the box and/or avoiding the math as much as we can.
What does that have to do with this example? Well let's think about what exactly happens to the ball after it is thrown. Initially it has a positive velocity and is slowly or quickly slowed by the earths gravity. Eventually the gravity wins and the ball falls back to the earth. But let's reexamine precisely what happens at the exact moment in time when the ball reaches it's apex, what would it look like if we could freeze time at that exact instant and examine what the ball is doing?
Well if you think about it, at that exact instant the ball is neither rising nor falling which means that the ball actually has zero velocity. This might sound incredible but think about it (and if needed I can try and reexplain this, it's been so long this concept is intuitive to me and so I struggle to explain something that is as evident to me as the fact that we breathe air).
So if we look at the equations we started with, we see that the 2nd equation gives us precisely what we need. As delta s is effectively s given that we have defined s0 to be zero and we know the initial velocity. From there it just becomes a matter of plugging in numbers. Remember though that when you plug in those numbers that acceleration is negative. Failing to do so will give you a negative height which should make alarm bells go off that you are doing something wrong.
The last topic I want to briefly touch on is motion in higher dimensions, now I am not going to go too much into this. Somewhat because linear motion explains a lot of problems and further that moving past linear motion is some what complicated. That said, essentially dealing with say 2 dimensional motion is not that complicated. All you really have to do (as acceleration is a vector) is break things down into their vector components. Thus you have to talk about acceleration in the x axis independent of the y axis.
Well, in truth that is what is preferred. Some physics though does not allow for the axises to be separated and that gets complicated fast. At that point you have moved into euler-lagrange territory. Which is beyond the scope of what I am trying to accomplish here.
Looking at the length of what I have written I believe I will save an example of 2 dimensional motion for next week and in fact will save a slightly more in depth treatment of the subject till then.
I hope everyone enjoyed this installment, for those that want to see the math behind the equations look for that on Monday or Tuesday.