https://youtu.be/8N8Z2VfgHMA
PHYS 1101: Lecture Four, Part Five
We’re ready now for our last category. We now have gone over specific ways of defining location for an object and defining its velocity. Both of those are vector quantities, so we need to define axes directions, and we need to define an origin for them to have a unique, specific meaning, these quantities.
The last quantity we need, is something that will indicate, very specifically, if an object is speeding up or slowing down and by how much. We’re going to learn as we move on to Chapter 3 that we also need something to tell us if the direction is changing, this object’s rounding a corner, going over a hill. All of these quantities are captured by what’s called the “acceleration vector”. The acceleration vector is going to tell us how much, and in what direction, the velocity vector changes.
Remember that the velocity vector told us how the position changed. If you remember the definition of it, the velocity had to do with the displacement, comparing two positions of the object. The acceleration just goes one step beyond that. It’s the next level of detail. The acceleration vector is going to involve the difference between two velocity vectors.
I have the same comment about the acceleration that I did about the velocity vector, meaning that, if you think about it, for us to say anything about how the velocity is changing, we necessarily need enough time information that we can compare two velocities. I can tell you if that speedometer is swinging to the right or swinging to the left or not changing, if the car is cruising along at a constant velocity. So that translates to, just a single glance at your speedometer isn’t enough. You need to compare the speedometer reading at two instances. I need to know how the velocity is changing as time goes on.
Okay. The acceleration vector does that for us, and here’s what it has to be. If we mathematically define it this specifically, it’ll do the job for us, it’ll capture this information with the exactness with which we need to know it.
Okay. This acceleration vector, we’re going to use a with an arrow over it to indicate it. It’s going to have a magnitude and a direction. It mathematically is going to be defined as a final or a later velocity minus an earlier velocity. Final minus initial, divided by a time interval. Okay. Another way to write this is change in velocity divided by our change in time, our Δt, time interval.
So let’s go down to this quantity again, down here, and because it’s a vector, I want to do the same thing I’ve done before. I want to clearly spell out for you what sets the magnitude of this vector, and where does it get its direction information, what sets the direction.
Okay. Value first. The value of this, the magnitude, comes from the numbers or the values on the right. It’s going to come from the value of the numerator, which would be the magnitude of this vector, divided by the value of the denominator. So magnitude comes from the magnitude of this vector Δv divided by the value of the time interval.
It’s going to have units of what? What would be the units of Δv? Well, you’ll see in a minute that Δv is the difference between two velocities. Any sum or difference has to have the same units as each of those terms. Units of velocity is meters per second. That’s what I have to have in the numerator, and the denominator is going to have units of seconds. Units of acceleration, the magnitude of this vector, is meters per second per second. Mathematically, that’s equivalent to meters per second squared.
Where does this vector gets its direction information? The only direction information I have on the right-hand side, which sets the direction on the left side of the equation, has to come from vector information on the right. The only vector I have is Δv, so the direction of Δv is the direction of a.
Let me summarize that for you. Direction is going to be the same as direction of this Δv vector. What is that direction? That’s the next thing we need to look at a little bit more carefully. It’s the vector that answers the question, “Is the velocity changing?” And here’s a good visual way of assessing what that is.
We know to determine Δv and acceleration, I’ve got to compare two velocity vectors. So imagine a motion diagram where I’ve got three snapshots next to each other and I’m going to compare this later velocity to the earlier velocity. For this object, you note that it must be speeding up and it’s headed to the right. This vector is larger than this one. So what is Δv between these two? What is the change?
Let’s first do a tail to tip analysis. Let’s go down here and I’m going to copy this final velocity vector. Let me draw that down there and make it a little smaller. I’m also going to go copy the initial velocity vector. I’m going to line it up here right beside it. It looks like I need to move it to the left just a bit.
What I want to show you is that the change in velocity, this Δv, physically means if I start with this velocity vector, what vector do I have to add to this, tail to tip addition, in order to equal or get the next velocity vector? Well, v0 gets me about halfway. I need to add a vector that’s this large in order for the tail to tip sum of v0 plus this new vector to be equal to my next velocity. So this vector that I drew, this is the change in velocity. This is my vector Δv, the change in velocity.
Let me just write mathematically what that says, then. This final velocity is equal to the initial velocity plus the change, plus this Δv vector that I had to add to this initial vector to get me my next one.
So I’ve summarized it for you down here. Let me put a star to try to emphasize that. The change in the velocity, as I go from one instant to the next, is this Δv. That arrow represents our acceleration because, remember, I pointed out that the direction of a comes from the direction of this Δv vector and I’ve just shown you that that Δv is going to point in the direction and be the vector that I need to add to the previous velocity vector. So, this little Δv arrow that I draw, when I look at this motion diagram, and up here I picture, what do I have to add to this to get me the next one? I need to add a small amount in the positive direction. When I do that, I can think of this Δv vector that I’ve just drawn as being equal to, or representing, the acceleration vector.
Okay. I just paused it to tidy this up here a little bit for you so your notes will be clearer. Let me just show another quick way of drawing this. In order to get roughly the right size or length for this Δv, which I can use to represent the acceleration between these two. I think of it as the acceleration, maybe at this instant in between is a good way to think about it. I can go in and let me imagine taking this vector and repeating it here starting at this next point.
Once I have it overlapped there, then it’s clearer to me that this is the vector I need to add to v0 in order to add up to my next velocity vector. Let me erase those points so I don’t mess up the picture too much. So repeat drawing v0 and then from the end of that, draw the vector you need to add to v0 to represent this next velocity.
Here’s another way of thinking about it. If I had no acceleration, if my velocity didn’t change from point to point, my velocity vector would be the same, but it isn’t the same. At this next point, I don’t have this. I have this plus this. So this thing that I have to add to show how that velocity is changing is a representation of my acceleration. It realistically is Δv and I can use that arrow to represent a because a is defined to be set by the direction of Δ v.
All right. That’s probably overkill on that. Let me erase those so it doesn’t muck up the drawing. We’ll practice that some more as the class goes on.
Another quiz question for you, I want to see if you physically know what acceleration means. Let me remind you that a good way to think about this is when you see an acceleration value like this, 3 meters per second squared, think about the units, and I encourage you to rewrite this number so the units are clearer.
Here’s what I mean. First of all, I know that this is acceleration because of the units, meters per second squared. Another way to write the units is to say that this is 3 meters per second per second. The acceleration tells me that every second that goes by, my velocity changes by plus 3 meters per second. That should help you answer this question.
Okay. I’m going to go through the same steps I’ve gone through with the previous quantities. I want to be sure it’s clear to you how we’re going to write this acceleration vector. As always, because it is a vector quantity, when we have defined our axis direction and the meaning of positive, the sign we’re going to associate with a is going to reflect how a points, how the vector points compared to our positive and negative directions.
Okay. Here’s what I mean. Here’s my scenario showing you this is the proper Δv to represent the change in velocity that I have for this scenario. Now, I’ve told you that because a is equal to this Δv vector divided by the time interval, this arrow I can use to represent a because that’s where a gets its direction information from, is from Δv. So this is how I’m going to usually draw our motion diagrams and add to them the acceleration vector.
Mathematically, this vector would be written as, again, a scalar component multiplied by the unit vector to tell us if this is a vector that’s horizontal or vertical and then the sign and the value of this will tell us what the length, the magnitude, is, what the length represents.
Okay. The summary I have down here is what I’ve said before. Just to remind you that this is a scalar component, sign is important, and it’s always going to represent the direction that this acceleration vector points. That’s the same direction of our Δv.
Okay. For your next two quiz questions, I’m going to again show you a short, little video, a short motion movie. We’re going to use the convention we’ve done before with our positive x-axis pointing to the right. With this sign convention, this direction to find, I want you to watch this first movie and tell me, approximately one second after the motion starts, about in the middle of the motion, what’s the proper sign for the acceleration vector?
Okay. Here’s the movie. The motion starts now here on the right and this object is moving to the left, so somewhere in the middle here, about one second after it starts, what’s the sign of the acceleration vector? To answer this, I would pause it and think of these dots as your motion diagram dots. I know my velocity vector has to point to the left. I know this velocity vector is a little bit larger than the previous velocity vector, so what’s my Δv vector? Does it have to point to the left or to the right?
Whatever decision you make there, that’s the same direction that a has to point. You then have to choose the sign of a based on the fact that a positive arrow would have to point to the right, or a positive scalar component means that vector points to the right.
Okay. That’s that question. And now for question 18, I want you to watch a similar movie and answer the same type of question. What’s the proper sign for a about the middle of the motion? Okay. This is the movie I want you to watch to answer question 18. This motion is different. The object is still going to the left, but now you notice that it’s slowing down. For this scenario, with the positive x-axis defined to be to the right, what’s the proper sign for the acceleration scalar component roughly here in the middle of the motion at one of those instances?
Okay. After thinking about that, it’s worth going back and appreciating, or thinking carefully about the sign that you picked for the acceleration for those two and how that agrees with, or disagrees with your intuition. I suspect that in that first movie that you were watching, where the speed was increasing, you really strongly want to pick that the acceleration scalar component must have a positive value, that that’s what increasing speed, or speeding up, would have to mean.
Likewise, for the second movie that you watched, where that object is slowing down, that you really wanted to make the acceleration negative, that that’s what decelerating would be. On one hand, I appreciate the common sense intuition of that very much, but you have to have the specific definitions of what the sign means for this class override your intuition. Okay. For this class, we’re mathematically using the tool of sign, not to mean decelerating or speeding up. The sign has to mean the direction of a vector. That’s what you have to go by to answer those previous questions. Okay. So note, counter-intuitive, must override definition in physics of what sign means must override intuition.
Okay. Here’s a quick mathematical discussion to show you why this is, why this sign has to convey direction in order for the mathematics to work out. Let me walk you through this so you see what I mean. Remember that our equation for a is defined this way. It equals the difference between two scalar components and the time. Note I am not using t0 here and that’s because we’re going to define the initial time to be equal to 0. That’s where we’re going to start our stopwatch, start our clock.
So given this definition, this is what our equation for a becomes. You’re looking at it in its scalar component form. The sign of this, this, and this represent the direction of these vectors in whatever direction our motion is, be it horizontal or perhaps vertical.
Let me take that equation for you and just rearrange it a little bit. I want to multiply both sides by t and I’m going to add v0 to both sides. My goal here is to look at an equation that says the final velocity is equal to what? And so I’m going to mathematically rearrange this so I can then translate that equation to see what, physically, this is. What sets our final velocity?
Let me rewrite the equation where I am so far. I’ve got a times t is equal to v minus v0. Well ,it looks like now, if I’m trying to isolate v, I need to add v0 to both sides. This then will have effectively moved v0 to the left side of my equation and now I’m going to swap the order here so I can read my equation from left to right in my equation.
Okay. So now, rearranging, I have that v equals v0 plus a time t. If I rearrange it like this, and I read it, the sign can start to make sense for how critical it is that the sign of a and v0 represent the direction of these vectors. This equation reads that the final velocity is set by, is equal to, the velocity I started with, plus the acceleration that I have, times time. So here I’ve written it larger for you and explicitly spelled it out, this interpretation or translation of that equation.
So when you write it this way, and you do this translation from the math back to real life, it’s easier to see that the sign is important and that it makes it all work out. First, let me point out that the only sign we’re ever going to have in this equation is going to be a positive or negative a for this scalar component or a positive or negative v0 for this component. Time is always positive. It doesn’t physically make sense to talk about a negative time. So whatever the sign ends up being when this is added together will set the sign of our final velocity. Physical translation, that’ll set the direction that we end up going after some time has gone by, after we’ve witnessed some amount of acceleration, given that we’ve started with some initial velocity.
So let’s compare the two scenarios. What if we want to represent speeding up? If we want our final velocity to have a larger magnitude, a larger value, we need to have the sign of these two terms be the same. If they’re both positive, adding a positive number to a positive number gets me a larger positive number. Physical interpretation, my velocity started out, say, going to the right, if I do a positive definition to the right, my acceleration, my Δv’s — I can write it both ways — my acceleration is to the right, my initial velocity is to the right, so if I add positive to another positive number, I get an even larger positive number.
This shows me that I’m actually going to be speeding up to the right. If I want motion that’s going to the left, then my initial velocity would be negative because it’s to the left. If I use a negative Δv, then I’m going to be adding a negative with a negative to end up with a velocity that is an even larger negative number, meaning that I have sped up, but I am still going to the left.
Okay. If you think through slowing down, you’ll realize that slowing down will be captured by these vectors having the opposite sign. That’s consistent with a motion diagram, for example, where the later velocity is smaller than the initial velocity. So to this velocity, I need to add a vector that points to the right. I need to subtract from this initial vector to end up with a shorter vector at the next instant. So this Δv, or this acceleration vector, points opposite if I want the thing to slow down, and that’s true if it’s going to the right or the left. I just have to have opposite signs for these two vectors.
So here’s our variable summary, as I’ve done all along the way. When we work these problems, we’re going to have to read the information and map what’s happening physically in this problem in the real world to our mathematical representation of that, our variables. We’re going to use the scalar component a to represent acceleration. We will always convert to, or have units of meters per second squared for acceleration. Another way to think about that is meters per second every second. This, I think, is a good way to interpret it to give you a physical picture of it. That velocity is changing by so many meters per second every second.
Okay. Let’s map that onto our problem that we’ve been visiting again throughout this lecture. We’ve got a scenario of a jet liner traveling northward. I’ve got it sketched again to the right. I’ve gone back to my initial axis definition direction, so we’re going to make positive point at the northern direction. So to answer this question, take these steps. Slow down and these will help you arrive at the correct answer for question 19.
First, on this, or off to side, do a quick sketch of the motion diagram. Is the spacing getting closer or further apart for this object? Once you sketch that, draw your velocity vectors in between those points. Go through that exercise of comparing two adjacent velocities and ask yourself what’s Δv? Do I have to stretch or compress a velocity to get to the next velocity to properly describe it? That arrow that you draw to represent what you have to add, either in the same direction or the opposite direction of the velocity, will tell you the arrow direction for a. Okay. Go through those steps and then answer question 19. What’s the right sign for a?