https://youtu.be/ALK5H93YnoQ
PHYS 1101: Lecture Six, Part Six
Okay, all those plots that we looked at above were fairly straight forward in that I had nice straight lines with different angles and slopes. So that’s a good place to start to see if you know how to calculate slopes and if you have a physical picture of what the motion is.
In real life, often the object’s position changes in a much more interest way. Curves are not always straight, they’re often curved. Whether it be the position versus time graph, or the velocity versus time graph, there’s some curvature to that. An important point, is that this notion of the slope of this curve telling us something about the velocity at a point is still valid, but now we’re using the ideas of instantaneous velocity, meaning at a certain instant in time, what’s the slope.
The mathematical term for that, when you have a curved line, is the “tangent line”. The slope of a curve at a particular point is called the tangent line. So at that instant, dash yourself a line up to where the curve is and we have to sketch the line that’s tangent. Once this red line, the tangent line, is sketched, forget about the blue one, just focus on the red, and say, “What’s the slope, what’s the rise over the run of this line, this straight line that you’ve drawn?” That then will be the instantaneous velocity.
So let me write here for you, that the slope of tangent line at some time, t, is the instantaneous velocity. So is v instantaneous? Let me write it better. Is the instantaneous velocity.
The velocity of this object undergoing this motion is increasing the whole time, it’s not constant. This object is accelerating. It’s velocity is getting larger and larger, because I can picture that the slope of a tangent line is getting steeper and steeper. See how the slopes are getting steeper and steeper.
How do you find that tangent line? It takes a little practice. Let me give you some guidance here at first. Let’s start out, and just draw a very big, exaggerated curved line. And let’s say you’re interested in the tangent line at this point. So this points corresponds with the particular instant in time, or t, that you care about and you need a tangent line here because the slope, once you draw that tangent, and you calculate the slope, that slope will be the velocity at that instant.
And of course this assumes, I’ve assumed here that we’re looking at a position versus time curve. Okay, here’s a suggestion to think about how to do it. What you’re really doing is you’re trying to find the slope of a line as you zoom into this region here.
What does “zoom in” mean? Imagine you start out quite far from that point, and you draw two points equal spaced and then you draw a straight line through them. I’m not sure that I can do that very well. Not bad. Okay, so now this line has a slope, but it’s not the tangent line yet. Continue doing that by picking . . . you’re zooming in. Pick two points on either side that are closer to the point and then draw a line. Continue doing that and whatever line you put your ruler down and draw a straight line between these two points, that the slope of that line is what you need. You’re converging to the tangent line.
Once you get the hang of it, you start to realize that all you’re really after . . . have some space below here, I don’t. All you’re really after is a line that, if you’re interested in this point here, it’s a straight line that’s sloped the same . . . I’m not saying that very well. I’m going to draw this out better. That’s not a good one either. If you zoom in here, I’m trying to say it’s a line that kind of has an equal angle that you see on either side of that tangent line.
So what’s the tangent line here? It would look something like that. What’s the tangent line here? Something like that. Yeah, practice that. You’ll need to be able to eyeball the tangent line.
So, here’s a quiz question for you, number 17 to see if you have the hang of that. It’s showing you the position versus time again for two objects, A and B, and asks you if you ever have the same speed. So as I’ve pointed out before, remember speed is the magnitude of velocity. What’s the velocity, if you’re looking at x versus t, you have to think slope.
This brings us to the end of lecture 6. I look forward to seeing you in the Sentra online help sessions, or perhaps in the resource center.