https://youtu.be/NxqWSoVOfNo
PHYS 1101: Lecture Five, Part Four
The next page, this section is a summary of my problem solving steps. There’s a copy of this page in your equation sheet document. That document, we’re going to keep adding to as the semester evolves, and you can always find the latest version of that by clicking on your chalkboard link on your homepage. It’s this link, this icon here. I just pasted it for you. You’ll see this at the homepage, under “Web CT” for our class.
When you click on that, the first few pages you’ll find conversion tables, which are on the front cover of the hard copy of the book, hard bound copy. I’ve copied those there for you in case you just purchased the online version of the textbook. But then, beyond those tables, the very first page is a summary, or it is this page of the kinematic problem solving steps. So if ever you’re stuck on a problem, go back and follow these steps. 99% of the time, it’ll unstick you. But, you got to know what these steps mean. That’s what I’m going to walk you through carefully as we do some examples.
First important point. Here are our fundamental equations that we can use. These equations, there’s assumptions behind these equations. We have to be sure our problem meets or satisfies these assumptions. These equations are only going to work to describe the motion of one object. That object can only be undergoing constant acceleration. It might be 0, in which case, these equations would customize, these terms that involve a would go to 0. So, the equations would become simpler. Or, I may have an a. It could be a real number, but it has to be the same constant number from the start to the finish.
Now, some of the problems are going to involve motion where I don’t have constant a from the very beginning to the very end. Like this example that I pointed out in our summary at the beginning of the lecture, if that’s the case, I still can work with these problems. I can still use those kinematic equations, but I have to break the problem up into parts. I can use those equations to solve for motion for the first part, a part during which a is constant. Then, though, I have to think of somewhat starting over with a fresh set of variables, a fresh set of equations to describe the motion for the second part, for which a has a different value.
I then may need to connect the two by noting some similarities or some common features. For example, at this instant in time, whatever the position was at the end of the first part, that necessarily has to be the same position as the start of the next part of the problem. We’re going to do an example like that. You’ll see what I mean.
Okay, so, here’s my reminder. One object, constant acceleration. So, these equations are what we have to work with. The name of the game is slowing down, reading the information slowly enough so that we can customize these equations to our problem. We can customize these variables and what their meaning is for our problem. The steps that I outline here help you do that.
Step one: visualize. Read the problem. In your mind, can you picture the motion as it’s being described? And what you’re supposed to focus on, you have to pick the object. What is the object you’re supposed to be looking at whose motion you want to describe? Clearly identify what’s the scope of the motion. What instant’s going to represent the start, what snapshot in time? And what snapshot’s going to represent the end? Again, you got to have constant a between the start and the end.
Draw a motion diagram for that. This is really, really helpful and important. That motion diagram, when you draw in your velocities, and then think about how those are changing, that will tell you the direction of the acceleration vector. That’s key for picking the right value and the meaning for the a that shows up in these equations.
In order for these vector quantities to have meaning, you know now that you have to define an axis. You have to know what the positive direction for vectors is going to mean, what direction that is, and you have to have an origin. The origin is the only way that these position vectors are going to have a unique meaning. When you say that the object ends up at plus 50 meters, you have to know where 0 is, or it has no meaning.
Define your axis. Define your origin. It’s up to you to pick that, and you have to be just consistent then throughout the rest of the problem. I’ll work a problem for you, where we change that, we change the origin, and then you’ll see that we get the same answer, as long as we’re consistent.
With that sketched, your origin and your axis, add to that drawing. Just clarify in your mind what the motion is, what the object is. Identify a start somewhere on your page and then an end. Sketch the object at that starting location. Here’s where your Pictionary skills will come into play. Doesn’t have to be fancy. Just put a dot there if you want. Sketch the object at the end.
At the start, I’ve got time, position, and velocity to think about. What are these vector quantities? Are they numbers given in the problem? Are they unknown? Are you supposed to solve for one of these?
At that initial position, go ahead and sketch an arrow to indicate the direction and the rough size of that initial velocity. At the end of the problem, sketch another vector, another arrow to represent the final velocity. At the end of the problem, just at that instant, what direction is the object headed? And roughly, how long should that arrow be compared to the initial velocity? Was it slowing down or speeding up?
At the end, as at the beginning, I’ve got to think about time, position, and velocity. These two are, again, the vector quantities. The sign of this represents being on the right side, the positive side, or the negative side of the origin. Sign of this means at that instant, the object was headed in the positive direction or the negative direction, etc. Remember what these variables physically mean.
Then I, somewhere on my page, in between where I’ve sketched the start and the finish, I draw a little arrow and label it ‘a‘ to remind myself of just the direction of the acceleration. That information you will know, having completed or done, either on paper or in your mind, a motion diagram so you have a picture of what a is.
The next steps are to make a list of these variables that you know, and what variable it is that you need to focus on and solve for. We have to map the problem into these 1, 2, 3, 4, 5, 6, 7 variables. Which of these do you have values for? You have to pick one of them as the variable to solve for. It may be the direct answer that the problem is asking you for, or it may be related to that, but you have to pick one of these variables if the equations are going to be useful, because the equations only relate these variables. You look at the variables that you have. You see what variable it is you need to focus on. We’re going to, then, use our mathematics to work to our solution.
Then, at the very end, it’s always good to go back and just think about your answer, see if it makes sense to you. Units are a great check. If the units of your answer, or as you try to plug in numbers to get the final answer, if the units aren’t working out, you’ve made an algebra mistake somewhere.
Then, does the value physically make sense for the problem? Is the building a reasonable height? Is it reasonable that it takes a ball 3 seconds to fall off the top of a building? Use your intuition to see if you think you’re even in the ballpark.