Wooh! We are so very close to having enough math under our belts to return to physics, where it is MUCH easier for me to make penis jokes.
The Velocity Problem
Imagine I’m driving away from your sister’s apartment. My speed leaving isn’t constant. I leave slowly at first in all-electric mode so she won’t hear. Once I’m out of earshot, I switch to traditional engine and gun it. So, the graph that would describe my speed would vary over time. Very likely, it’d start slow then get faster and faster.
Now, say you want to know my speed at a particular point. That speed at a particular point is called “instantaneous velocity,” which we discussed a little back here. Let’s also say for simplicity’s sake that we have a simple equation:
, where p is position and t is time.
So, at zero minutes, I’m at your sister’s house. At 1 minute, I’m 10 meters away away. At 10 minutes, I’m 1 km away, etc.
Say you want to know my speed at a particular time. One way you can do this is by picking two points, as we did in the last segment, and making a right triangle with them. The slope of the hypotenuse will give you my change in position (p) over my change in time (t). Well, what’s change in position over time? It’s velocity!
But, you may notice a slight problem. You’re still not getting my velocity at a particular point. You’re getting how much it changed from one point to another. Obviously this could create problems. For example, say you were looking at y=sin(x) and the two points you picked were a trough and a crest. It’d look like I was leaving your sister’s house, when in fact I was leaving then returning then leaving then returning.
So what you could do is bring the points closer and closer to each other. No matter where you started, as you move one point closer to the other, you get a more and more accurate picture of my speed at a particular time.
Let’s make this all mathy. For points A and B.
Simply put: the average velocity over the time between A and B is given by the quotient of the difference in position (y axis) and time (x axis).
Now, similar to what we did in the last section, imagine you move B slowly closer and closer to A. In this case, let’s say A happens when x = 5. If you draw or plot that curve and move closer, you’ll notice that slope steadily approaches 100. That is, it really looks like the actual velocity at exactly 5 minutes is 100 mph. That value is the instantaneous velocity. It’s your speed at a particular point in time (an instant!), as opposed to a span of time.
Hopefully you’re getting a sense of what calc allows you to do now. Before calc, all you can do is get the average velocity. Using the methods in this and the last blog, you should see that you can get a lot more data from the same equation if you know the magic of Calculus. We’re only 6 sections from the apparently dreaded Derivative. That’s when things are gonna get really crazy.
Until next time!