https://youtu.be/pEWF0xKHz28
PHYS 1101: Lecture Seven, Part Seven
Now I want to show you an animation to give you a couple of animations, to give you a feeling for why this is valid, why it makes sense that you really can think of two-dimensional motion as being equivalent to two cases of 1D motion. You have a horizontal part and a vertical part. The only key, or trick, is that the horizontal part and the vertical part just are occurring at the same time and that by viewing it this way, stripping out the horizontal and vertical part happening at the same time, that it really does match real life and what your eye is drawn to, which is the real motion of the ball of the object.
So watch this animation, and I want you to focus on the purple ball. So here in the center is a real curved trajectory of a ball, following this curved arc with that purple ball. So that’s the 2D trajectory. Okay. I’ve got a sketch of it here for you to emphasize that. It’s the purple ball that’s the motion you want to focus on here. The trajectory of the object.
What was happening at the same time with the red ball and the purple ball was just an echo of where the purple ball was vertically at each instant in time and where it was horizontally.
Said another way, let’s imagine as we watch it again that we cast a shadow, a bright light, toward the y-axis, this vertical axis. By the shadow that gets cast from this purple ball at these, say, one second intervals, we will be seeing a motion diagram of the vertical part only.
Let me play it again for you. Okay. So I’m going to pause it here early on. So we see that these shadows are left at equal time intervals and at every shadow location you’re seeing, at each spot on our motion diagram you’re seeing where the y part, or the vertical part, of that object is, the y shadow. This one, you’ll notice, is perfectly aligned to this spot.
So if I go back to this picture, then, and I put my black dots here to represent the motion diagram for the vertical part to the motion, I see that the velocity is getting smaller as the ball goes up. The vertical velocity.
Let’s do the same now with the horizontal shadow. Let’s imagine shining a light down as we let that motion play. And we’ll let it go a little longer this time. So for these equal time intervals here, at the same time that the vertical velocity is getting smaller, smaller, turns around, and then starts speeding up, the horizontal part to the velocity, you’ll notice, doesn’t change.
The horizontal position of each of these purple dots doesn’t change. If you were to look at this kind of motion, fill in with my black dots to make it a bona fide motion diagram, you would call the horizontal part a constant velocity motion. The arrows are all the same length, pointing in the same direction.
So for the quiz questions that follow, we’re going to define standard axes of x and y, positive y up, positive x to the right. The first question for you. Here’s a blow-up of the vertical part to the motion. Here’s my motion diagram. If you focus on the vertical part to the motion, on the way up, what is [a sub y]? A sub y has to tell you how v y changes. It’s the delta-v y, it’s the difference between y velocity vector components, and that’s what these are. V zero y, and v y, the difference between these two.
After reaching the top, on the way down, are y, these red shadows, the y motion, follow this type of a trend? Our velocity was down and getting, the speed was increasing. On the way down what’s a sub y? What’s the change in the y part to the velocity?
Question 10. What is the y part to the velocity at the highest point? Here, at the highest point. So here’s a summary for you. If you go back and you think about the motion that you observed, Ay is negative and it’s the same constant amount all the time. If you look at the real difference between these vectors, that’s what you’ll discover.
Okay. Now I want to show you this movie again, but at the same time that this projectile, this object, is following this motion, curved path, rather than look at the motion diagrams on the ground and on the vertical wall here, those shadows, let’s look at the corresponding graphs that go with it for the x part to the velocity as time goes on and the y part to the velocity as time goes on. Let’s play the movie.
So this is the ball that we’re following, it’s following this path, and we have already asserted and realized that the horizontal part to the motion represents constant velocity. The shadow cast here, equal spacing, tells us that Vx is constant. Here’s my total velocity, the x component, and here’s the y component, and if I do that for every point between this motion diagram I will see that the x part to that vector is always the same length. That’s what our plot shows, too. This is a plot of Vx versus time. As the clock ticks on for every second, our velocity vector is always the same constant 30 meters per second the whole time.
Now, I want you to think about the y velocity graph. This is the graph that goes with that movie that I just showed you. Quiz question 11 is: at what time does the ball reach its highest point? Looking at this graph, can you conclude that?
My next question is, showing you again the x motion diagram, showing you it explicitly, let me dot in the positions and our Vx velocity vectors. Question 12 is, what is Ax? Remembering that Ax tells you what the change in the x part of the velocity is.
The next quiz question shows you my sketch of what the x was for this motion. Remember, it was at 30 meters per second. With the x at this constant 30 meters per second, my question 13 to you is: if you were looking at this plot alone or if you were looking at this motion diagram alone, can you tell when the ball reaches its highest point? And I see some excellent spelling there. Let me fix that up. Okay. Can you tell, A yes, B no if you look just at this?
Now I want to show you the same trajectory of this motion but I’m going to turn on, showing you the vectors, the acceleration and the velocity vectors, for the motion of that purple ball. Okay, I’m going to play it again. Unfortunately they’ve chosen purple to represent a velocity vector when we all know that green is the right choice. Pretend this is green. Acceleration they have properly drawn as red.
Let’s play it. I’m going to pause it here for a minute. The velocity vector always points in the direction the object is headed at that instant. The horizontal and the vertical components represent the v x and the v y. Notice throughout the whole motion that the acceleration is straight down. The total delta-v that causes this velocity to curve and arc around is, follow that motion, is caused only by a vertical acceleration. I only need Ay to get this curved motion. I only have my velocity in the y direction being impacted or changed.
The x acceleration is zero, there’s no x component to this vector, the acceleration is just straight down, and it’s that combination that gets me this gently arcing trajectory. Let me reset and play the whole thing for you. Notice the velocity vector, it’s always following the motion. The acceleration vector is always straight down.
Let me pause it at the highest point. Oh, I went a little bit past. At this highest point, I need to reset it. Let’s try again. Let me get there by gently stepping it. One more. Ah, perfect. At this highest point my velocity vector points in the direction that that object is headed just at that instant, if you picture looking straight ahead of you, if you were on a cart, on a roller coaster or something following this path, at the highest point you would be looking straight toward the horizon.
Once you crest the hill, then you start looking down. That’s the direction the velocity vector always points. At this highest point, this vector has only a horizontal component: Vx. There is zero Vy. That’s consistent with what we know. When we throw a ball up at the highest point, its velocity is zero. When it’s 2D motion like this, it’s only part of the velocity that’s zero, only the vertical part.
So here sits Vx, and notice the length of the x. You’ll find that it should be the length of Vx the whole time. Okay. Let’s finish that, and let it play out. Okay. So this motion that you’re watching is a result of starting out with an initial velocity at an angle, an acceleration vector that’s straight down, and I’ve showed you at the beginning of the lecture that that combination gives us an arc trajectory.