https://youtu.be/TtboO88tIrE
PHYS 1101: Lecture Five, Part Three
There is question 3, so you can see them both clearly.
So to begin our new content, we now have these tools of specifically defining location, velocity, and acceleration. If you have read those sections, you’ve seen that working with those definitions only, one can derive, mathematically rearrange these equations to come up with versions of those relationships which are very interesting and very useful to us.
The three equations that we are going to work with are summarized here. I am going to translate each of these for you carefully, so you have a physical feeling for it. I am going to give you some problem solving steps and how to use them in just a minute.
Before that though, I wanted to show you the big picture of where these equations fit in with our overall strategy for the class, of what we are trying to focus on and learn as we work through this material.
Remember, I emphasized on lecture 1 that our goal in this class is not to memorize the answer to a whole collections of problems, but rather to learn the fundamental physics behind these types of problems, learn how to solve these problems on our own.
I have my analogy here to an oak tree to emphasize that for you. The idea was that the basic physics really is at the bottom of this tree, and there are only a few concepts that we are going to learn. One of them now is the idea of describing motion, what’s called “kinematics” and these kinematic very general equations that can describe all kinds of motion. These are just for motion along a straight line, 1D, which is what we started with.
With these fundamental starting equations there’s going to be a whole collection of problems, for which these fundamental ideas are going to allow us to solve, or to draw some conclusions about the motion. So the picture, again, was that each leaf out on the end of this tree would represent a specific problem. A dog is chasing a cat, starts from rest and speeds up, how long does it take the dog to catch the cat? Or a ball is being dropped from the top of the building. How long does it take to reach the bottom? How fast is it going just before it hits the ground? A whole assortment of problems like that.
They can appear to be very different on one hand, yet we are going to see that the fundamental physics behind it is similar between all these problems, and they all have, at the heart of them, these very basic equations. So the question is for a specific problem, dropping a ball off the top of a building, how do we get from the fundamental equations behind it, the fundamental physics, out to this solution to this problem.
So you remember that it’s this process, these problem solving steps, that we’re focusing on in this class. And this is what we have been doing so far in the class, whether you have appreciated it or not. We have been going through and understanding some of the mathematical tools, some of the definitions that we need to do this problem solving process.
Every time we have a new starting fundamental principle that we’ve learned, a new tool that allows us to solve a specific collection of problems, I’m going to introduce for you problem solving steps, very specific steps to follow that will just help remind you of the kinds of things you need to think about as you make these choices and decisions to develop the solution to a particular problem. These steps, if ever you’re stuck on a problem, will help you. So I encourage you to keep those steps handy, and I am going to emphasize as I work through examples, of how you can use these problem solving steps.
So that’s what we’re doing for this lecture. I’m going to tell you a little bit about these equations, give you a physical feeling for them. I’m going to introduce you to these problem solving steps, and then we’re just going to work examples.
So at the base of that tree, the fundamental physics and the tools that we now have to solve a whole collection of 1D kinematic problems, that means one dimensional description of motion problems. We have really just three very useful equations to work with. I use and prefer these three fundamental equations. This is my version of what is Table 2.1 in your book. I want to go through each of these equations and translate the mathematics of the equations into what it means physically in real life.
The first point. These equations are a natural consequence only of the definition of what position vectors are, position coordinates, initially and finally, what velocity means, it’s Δx over Δt. The physical meaning and definition of acceleration, Δv over Δt. Simply given those definitions, if you have a case where the acceleration perhaps is 0 or a constant value, these expressions logically follow. These physical relationships between these variable have to be true. Let’s translate them one at a time.
Equation 1, it says that the velocity at a later time will always be equal to the velocity you started out with plus the product of the acceleration times time.
Let’s think about some scenarios here. Let’s say we have no acceleration. a is 0, this term, 0 times any time would be 0, and this equation would physically tell you what must be happening. If you don’t have acceleration, your velocity at a later time has to equal what it was at an earlier time. It’s the same the whole problem long, at any instant.
The next equation tells you about position, the position coordinate at a later time. It will always be equal to the position you started out with plus the initial velocity times time and then you have to also add one-half times the acceleration times time squared.
This is the proper equation that will tell you where the object ended up. Notice it relates position to initial values, time, how long the clock has been ticking since we were at this initial position with this initial velocity, and then I have this term here that involves the acceleration.
The last equation. It’s a very useful one in that it doesn’t involve the time. So regardless of the time, it tells you that the final velocity squared has to be equal to this combination. It’s the initial velocity squared, but then you have to add to that 2 times the acceleration, and then you’ve got to multiply by the displacement, the change in position, x minus x0.
I am going to emphasize it here and say it again below. I am going to get my big pen. Recall that these quantities are vectors. These variables are the vectors we’ve been talking about all of last lecture. The sign has to indicate the direction of these vectors.
The only exception to this vector conclusion or summary is that the variable time is not a vector. Time is always just a positive quantity. For these equations we are going to assume that the clock starts t0 is when we start the clock, so that corresponds to 0 time. So the t in these equations is always going to be so many seconds later in the motion, 5 seconds later, 10 seconds later, whatever it may be for that problem. This is the final time. These quantities are vectors except t.
Here is my summary of the variables. If you look at these three equations there are only six unique variables. At the start, I have variables for time, position, velocity. At the finish, at any instant in time I have variables to represent time, position, and velocity at that instant, at the start time, position, velocity. The only reason that t0 doesn’t show up in these equations is because we set it to 0, so that it would drop out, it wouldn’t show up there.
At every instant I have time, position, velocity, and then between the two, so I kind of draw it schematically like this with an arrow visually spanning this time duration of my motion I am watching from start to finish. I have to have constant acceleration. It could be 0, in which case my velocity is not changing, an object is moving along at a straight constant pace, straight line, or a, let me write here, equals a constant, e.g., a might be equal to -5 meters per second every second. That would be the constant value continually through the duration from start to finish.
So how many variables do you have total? You will always have 3 at the start, 3 at the finish, plus 1, the acceleration, in between. The name of the game is making our problem or translating our problem into these variables. I have to leverage these variables and match them to the information in my problem if I am going to leverage the strength of this mathematics, these equations to solve for unknown quantities to, solve the problem.
Here is a quiz question for you. For solving 1D kinematic problems with constant a, that is what we have been talking about, how many variables are needed? Even though you don’t see t0 in those 3 fundamental equations, go ahead and count it. So that is 1 variable. How many more are there? What are the total number of variables you need to consider or think about?