https://youtu.be/v9Vxxu_ll7Y
PHYS 1101: Lecture Seven, Part Three
2D Motion Diagrams
So the new material starts with a discussion of motion diagrams again, but now in two dimensions. The only trick to that is in two dimensions, that means our motion isn’t a straight line, but can follow a curved path. It can be any curve, just not a straight line.
So how do we generate motion diagrams in 2D? Well, certainly the dots that we put down and the velocity vectors that we draw between all the dots are the same notion we’ve had before. We would add to our, the picture of what we’re seeing, our sketch of it. We add the dots of where the object was at equal time intervals. And then between those dots, we add our velocity vectors.
Let me just turn one of these into a motion diagram for you. So that part still holds. Now we’re going to see, though, that the dots aren’t following a straight line, but they can follow it in the arc.
The real question we have to think a little bit harder about is acceleration. What is the acceleration vector once our path curves? What is a, which is the same thing as asking, again, what is the, what is delta-v, the difference vector, the difference in the two velocities, two subsequent spots.
So let’s start out with a simple scenario of having three points on a motion diagram, any three, for which of course I can draw the velocity vectors between them, and ask ourselves how to we get a from this motion diagram?
I’ve got some steps outlined here for you to help you do that. You’ll need to do this, we need to know what direction this acceleration vector points in order to understand, oh, so much of the future material. Forces, the future motion of the object, etcetera.
So here’s how you do it. Focus on any three points in a motion diagram. I have to look at least at that, because acceleration tells me how the velocity changes.
So, here’s what you want to do. I’m going to copy this over here first so I can follow these steps. Step one, at the same, draw the same initial velocity, the previous velocity at the next point. I’m going to do that with a bright blue pen. In fact I’m going to dash it in, draw, copy it here. And I’m going to change its color to blue.
So you can see what I’ve done. I’ve gotten, I’ve made an exact copy of v0 and I’ve just dragged it over to the next point. What this means, represents, is the scenario where I have zero acceleration. Zero acceleration means my velocity doesn’t change. So if I had this velocity at one instant, that’s what it’s going to be the next instant, and the next instant, and the next instant. But we see that the motion didn’t continue following a straight line, but rather it curved.
My next velocity is off at this angle, if I simply draw a vector from where the velocity would’ve ended up from the end of this v0 that I’ve just drawn to the tip of the actual v I ended up with, I will have sketched the vector delta-v.
Let’s think about that tail to tip analysis. V0 tail to tip plus vector delta-v equals, from the initial tail to the final tip, the new velocity. So this vector plus this vector equals the next one. I’m going to write it out and translate it for you, reading left to right. The latter velocity, v, is equal to the vector sum of v0 plus delta-v.
And this is the meaning of delta-v, my later velocity is what I started with, plus the small change. So, this vector that I’ve drawn, we’ve come to realize I can think of this as my acceleration arrow, my acceleration vector. So, from this point to this changing velocity, this represents the acceleration that that object underwent.
Okay and that’s what’s highlighted here in these steps, one, two, and three. Now a good way to draw this is to just slide this acceleration vector over and have it reside here in between these two velocity vectors. That’s just a good visual to show you that I start with v0 and then I have to add this acceleration vector in order to get the next v.
Let me tell you what I mean here. I’m just going to copy this down here and I’m going to take this acceleration vector that I arrived at from doing my steps here, from creating this mathematical equation which defines what delta-v is, or shows me what a is. Let me erase that black point there and just jot down that this is my acceleration vector.
And this is a natural place to put it, because it’s really in between these two velocities, where it’s natural to appreciate the change. I start here, with this velocity, here’s my change. And then I get to the next velocity. And you can physically think of it as v not undergoing a tilt, or a change in that direction to cause it to bend around and be more up, rather than down and to the right. I kind of physically interpreted it as, rather than v continuing on, I have to curve it up in this direction to get the behavior that I have the next velocity. I don’t know if that helps, but that’s how I think of it.
Practicing 2D Motion Diagrams
So now I want you to go through this exercise on your own and practice a bit this, picturing a motion diagram that’s following an arc and then looking at the difference between two adjacent velocity vectors to get the acceleration.
Do this while you’re picturing the following motion. Imagine your friend is swinging a ball around on the end of a string. And they’re swinging it around somewhat gradually and it’s following a vertical, circular path. Meaning as you look from the side, it’s like tracing out a circle, like a hands of a big clock.
Well let’s imagine the person is swinging the ball around in the counter-clockwise direction, and the ball’s moving at a steady pace. So draw this motion diagram, using about eight points around the circle, start at about 12 o’clock. So here’s my motion diagram for that scenario.
For Question 4, I want you, in text, using words, to describe the features of mine that you see. Talk specifically about the spacing between the points, and how that matches what’s physically happening and comment on the velocity vectors that I’ve drawn, their direction, size, etcetera. You only need a couple of sentences to do this. That’s Question 4 for your quiz.
Now for Question 5, I want you to carry out the same vector sketching and exercise that I did above, to figure out at point three here, so you’re comparing this velocity vector to this one, what best represents the acceleration? By saying point three, I mean using, considering this as v0 and considering this as v.
So go back, follow what I did above again, and now apply that to these two vectors. And then at this middle point, what’s the direction of the acceleration vector of the delta-v vector that you would draw? Here’s a copy of the points, or the steps again, to help you do that.