https://youtu.be/orca39RpLd0
PHYS 1101: Lecture Six, Part Two
Our next stage in the lecture is to give a summary of where we are, what we covered in Lecture 5, the key points. Lecture 5, we put together the main ideas of the variables and the equations that we use and can use to solve a problem about one dimensional motion.
The key points to that analysis, the key assumptions behind it, meaning what has to be true in order for us to apply these equations are described by these three bullets. The first bullet here is that this analysis, this problem solving approach we’ve learned, it describes the motion between what we would call the start of this motion that the problem is describing that we’re picturing and some finish instant in time. So over some time duration, we’re watching the motion of a single object, and during that entire time interval the acceleration for that object has to be 0, in which case its velocity would be constant, or the acceleration has to maintain a constant value. This would mean from start to finish.
For example, if I have an a of minus 2 meters per second per second, then from start to finish every second that ticks by, I have to start with that initial velocity, and I have to subtract 2 meters per second from it for every subsequent second. That has to be the case for the entire problem. If these three criteria apply, then these three main equations, quite generically, will always describe the motion for this object.
Now, each problem will be different in that our variables that we know, the features of the motion that we’re given in the problem, we know will vary, and the quantity we’re trying to determine will vary. It could be the final distance that the object reaches or a final velocity or how long it took to go from this position to that, etc. Whatever that problem, these generic equations still govern the physical relationship between these variables.
So these are what we can use as our mathematical tool to solve for the motion. The name of the game is to customize these equations to our particular problem, meaning we identify the values of these variables, as many as we can for our problem. We identify what variable we want to focus on to answer the question or to provide the solution. And then we algebraically work with the equations, manipulate them, ultimately trying to solve for or determine what the variable is.
Let’s say it’s acceleration we had to determine. a is equal to some set of terms or relationships that our algebra has yielded. That equation then is the translation to the physical answer that we need: a will be equal to . . . Once we plug in all those numbers, it will provide us with the correct answer.
Okay. If these equations are the tools that we have to use, then we have to be able to map our problem into these variables, into the physical means that they’re supposed to represent. I asked a question last lecture about how many variables were there? There’s a total of six unique variables that show up in these equations, and if you include the t0, what the clock reads when the time starts, when the problem starts, we’re going to use in these equations, in fact, we’ve already assumed that that initial time is 0. Count that variable and all the rest, you arrive at seven.
I emphasize this number because it’s a good strategy when you try to solve the problem. You’ve got time, position, and velocity at the start. That’s three. Time, position, velocity at the finish, that gives you six. Then you have to include the variable a, the acceleration, that would be between the two. If you make yourself a list of those seven variables and then go through them one by one and carefully look at the problem and see if you have that information, if you know that value, or that’s the variable you need to solve for, I think that will help you focus to solve the problems.
Okay, I’m going to make another comment here for us about the sign convention that we have because, remember, those seven variables, the majority of them are vectors. They are scalar components. That means the sign has to represent the direction of the vector for that quantity. The only exception to that is the final time. That’s just a scalar quantity and number, some positive number.
With that in mind, I want to show you a couple of scenarios. Let me show you, based on a convention, we’re going to use the standard definition here of a positive x-axis pointing to the right. With this definition, this mathematical convention, the direction of these vectors has to reflect the direction that they point based on this sign convention. Any vector then pointing to the right will be positive. A vector pointing to the left has to be negative.
So here I show you two scenarios. Here is an object that is speeding up, moving to the right, and we now know that a and v are parallel, if this is the case, because my Δv has to be in the same direction always to stretch and to increase that velocity vector. And here’s the scenario where the object is slowing down. My Δv, my acceleration, has to point opposite in order to compress that velocity vector.
So my question to you, and I’ll give everyone 3 points just for answering, I just want your opinion here. Given these scenarios, this object speeding up and assigning it a positive value of a, this object slowing down and assigning a negative to the quantity a, does the sign of those a‘s match your day-to-day expectation?
My next question is showing you similar but really an opposite scenario. All we’re doing now is contrasting an object against speeding up and slowing down, but now it just happens to be an object that’s moving to the left. Ideally, our mathematics, our ability to be able to describe motion should just as easily apply to objects traveling to the left and to the right. We don’t want to be restricted to just objects moving from left to right.
So, in this scenario, if it’s speeding up, I know my Δv has to be in the same direction. That means it has to point in the negative direction, same signs give me speeding up. For the case where this object is slowing down, Δv has to point opposite. Opposite arrows, opposite signs.
My next question to you, again, for 3 points, just give me your opinion, when you look at this scenario with a negative acceleration assigned to an object that’s actually speeding up and a positive acceleration assigned to an object that’s slowing down, do those signs match your day-to-day expectation?
The point I want to emphasize between these two and it, again, goes back to wanting to adopt the meaning of the sign as being the direction of this vector and that the strength of that is because we can then describe objects that are moving to the left or to the right and have the mathematics work out for us. I just want you to answer these two questions and kind of face that issue in your mind, which is probably, what my suspicion is, what disturbs a lot of people about the signs for these quantities that we have to choose.
The last topic to summarize from the previous lecture is the focus on the acceleration at a turning point, and here is my big bullet that I’ve highlighted before. I’ll say it again, it’s never 0. It can’t be 0. If the velocity just before and just after a turn, if it exists, it was headed one way, and then all of a sudden it’s turned around. The Δv, the difference, before and after that velocity it’s big. It’s not 0. I may have velocity 0 at the top at that instant, but the acceleration, the change, how it’s changing at that point isn’t 0. It always points to the center of the turn.
So here I show different scenarios. Here’s one perhaps involving throwing up and then coming back down. The acceleration at the top points to the center of the turn. Here’s one. Perhaps a ball rolling up a hill, it turns around and comes back. Again, acceleration points to the center of the turn.
Here’s one, an object going up. Who knows? This is a ramp of some kind and then coming straight back down. Whatever the object is, whatever that motion is, I know it turns around at that point. Acceleration to the center, always.
At the bottom here I just want to show you a bunch of some examples. This is some object that swings to the left and then turns around and moves to the right. At the end there it stops briefly but has to have an acceleration to the center. Same thing here. Now it’s moving to the right, swinging around to the left. And here’s another one rolling up a different hill now, perhaps rolling up the hill to the right, turning around and rolling back down.
The direction of these vectors always matches real life. It has nothing to do at this point with an axis direction that you’ve chosen. This shows you real life. This is the real direction that that object was headed, to the right, the relative length here. It gives you a feel for how fast it’s going, and these red arrows are acceleration vectors. They physically do have to match and represent whether the object is speeding up or whether it’s slowing down.
Once you only assign a positive or negative sign to these scalar quantities for these, then after you’ve defined what your axis would be. So for this object here, let’s say that it started out going fast, perhaps, to the left, slowed down and turned around. If in solving a problem about this kind of motion, I went with the standard positive horizontal axis to the right, then and only then can I go ahead and say, “Well, you know what? a, the scalar quantity, has to be a positive number for all of these cases.” The acceleration is constant here.
The velocities on the way up are to the left, so these v‘s now become negative numbers, but on the way back these v‘s now would become positive numbers. Okay, I hope that clears it up for you, helps you out.
The last thing in our “Where Were We” is a copy, again, of the problem solving steps that I’ve given you to help walk you through solving these one-dimensional kinematic problems. Kinematic, again, just means describing motion, what’s its final velocity, how far did it go, what was its initial velocity. Whatever that may be, it’s just describing the motion.
And, again, here is our picture that at the base of this big oak tree is these generic fundamental equations that apply to all problems that may be moving along a straight line, have a constant acceleration. The question is how do we customize or apply these general equations to solve a specific problem, a specific leaf, if you will, out at the end of this tree.
And so these steps are helping to guide you through moving up this tree to arrive at your specific solution, and the name of the game is, given the variables that are in these equations — there’s seven of them – usually given that t0 is 0. Leaving you six others that you for sure need to think about. What are those values for my problem? You must know. Memorize. Make yourself flash cards, whatever it takes. You have to know physically what each of these represents and be able to identify it properly in a real life problem, in real life.
These are vector quantities, meaning the sign is going to be important. What defines the sign? You have to do a sketch of the problem. You have to identify an origin and an axis direction. And all of those points that help remind you of that are summarized here in these steps.
So here they are, again, the variables, one, two, three, four, five, six, seven. As you reach the step and you go through these, just take them one at a time and ask yourself, “Which of these do you know? Which are you given? What do you have to solve for?”