https://youtu.be/7w_vZW0tZkM
PHYS 1101: Lecture Four, Part Six
What’s the right sign for a?
I have one last point for this lecture to emphasize to you or to go over. Warning, warning. This is a counterintuitive point that most people start out thinking about incorrectly and then if you slow down and follow the steps I’m going to walk you through, you’ll realize what the right interpretation is.
Okay, this starts with us watching a video that we’ve watched before with a ball rolling up a hill. But, this time we’re going to watch it roll up and roll down the hill.
So, here’s the video I want you to watch again, the short little movie. You’ve seen it before, it’s a ball rolling up a hill. This time I’m going to let it go and you’re going to see it roll up and back down. It rolls up to the top. Can you see it briefly comes to a stop? And just an instant turns around and then heads back down.
We’re going to think about velocity and acceleration and what our motion diagram tells us for this kind of motion. It’s what I want you to walk through carefully.
So here are the first two questions I want you to think about with respect to that movie. On the way up, the ball is slowing down. Does that mean that the acceleration vector has to point opposite to the velocity or the same direction as the velocity?
Let me bring it back here, play it again. Noting, the axis direction has been defined for us in terms of sign. But, here it’s just asking not specifically about sign just yet, but rather when you watch this movie, the velocity on the way up is definitely to the left. Is the acceleration vector the same direction as that or the opposite?
The next question, 21, is on the way down. Definitely the ball is speeding up. I keep playing it here, as you see, speeding up. Does that mean a has to point opposite to the velocity or the same direction as the velocity?
Question 22 is after you think about that you might want to do a little sketch to draw the velocity vector directions and then to draw on the way up and the way down the corresponding velocity and then draw the acceleration. How does the direction of your acceleration vector on the way up compare to the direction of a for the way down. Is it the same or is it opposite?
Okay. Go back if you like. Pause it, watch that movie again. I tell you what, I’m going to play it for you again. Let’s watch it. Very mesmerizing. Definitely is headed to the left on the way up. Negative velocity direction. It’s slowing down. It then stops briefly, turns around. Once it’s headed to the right it now has a positive velocity, and now it’s speeding up.
So, let’s think about the motion diagram for it. Now, for the complete motion diagram, of course my dots on the way down would overlap my dots on the way up, so I’m going to offset it a bit just for clarity. Here’s what I would draw for it. I’ve got indicated here the start and I know on the way up my velocity is pointing in the negative x direction to the left. That object is slowing down. This velocity vector is shorter than this one. My Δv has to be to the right. This has to be the direction of Δv and therefore of a. If something is slowing down, a has to point opposite to v.
Okay, after I get to the top and I turn around, I’m on my way back, the velocity now is to the right. Now the subsequent velocities are getting larger, the thing is speeding up. In order to represent that or have that be the case my Δv has to point in the same direction in order to stretch that vector every instant. So, a has to point same direction as v.
I want to focus, though, at the very top and ask you a couple of questions about what’s happening there. First question 23. At that instant that it comes to the very top, what is the instantaneous velocity just at that instant? Is it positive, negative, or 0?
The next question for you, is what’s my Δv right at the top? What’s the acceleration at the top?