Section 2.2: Instantaneous Velocity
In this section we’re going to touch on the mathematical concept of derivatives. We won’t get into a rigorous sense of it until quite a few more Calc sections, but this is a good opportunity to get an intuitive sense.
By “instantaneous velocity” we simply mean the velocity at a particular point in time. Note, we don’t mean the velocity over a very small period of time – we mean the velocity at a single point.
More rigorously, the instantaneous velocity represents how position is changing as time changes at a particular time.
Recall in the last section we talked about , where the delta refers to change. That fraction is called “average velocity” because it’s referring to the change in position over a certain amount of time. Note the difference: Average velocity is the change in position over a span of time. Instantaneous velocity is how position changes with respect to time at a particular instant.
Another way to think of it is using the mathematical concept of limits. Once again, I’ll save the rigor for the calc blogs. Here’s the basic idea: Imagine you start with the average velocity equation. That is, you say “position will change by a certain amount as time changes by a certain amount.” Now imagine you start shrinking the amount of time change we’re talking about, slowly zeroing in on a particular point in time. If you keep doing this, the time span in question approaches 0.
A limit is simply what you get as you bring that time span closer and closer to zero. Thus, we say “Take the limit of as approaches 0. When we do so, we get the instantaneous velocity.
We denote the instantaneous velocity simply as v. But, what’s under the hood is something a bit more meaningful: . The d refers to the instantaneous change. So, when we say dx/dt we’re referring to how position changes with time at a particular instant.
Allow me a quick digression: The dx/dt notation is a little weird since it’s better about showing you what you’ve got than what you’re doing. A better way to say it might be this: You have an operation “d/dt.” You can run it on things, such as x. What the operation does is it asks a variable how it changes with time.
So, say you’re in a car. You know a number of facts about how things change in this car over time. You know speed, but perhaps you also know temperature (T) and brightness (B) and how much the car smells like the creamer rotting under the seat (C). Using your d/dt operation, you can ask the instantaneous temperature change, dT/dt, or the instantaneous birghtness change (dB/dt) or the instantaneous stankness change, dC/dt. So, when you see something like dx/dt, don’t just think of it as a ratio. It’s more than that. It’s a question you ask of x.
Additionally, it doesn’t have to be d/dt (instantaneous change of something per change in time). It could be per change in height or change in size or change in stankness. dt is just very common, since when we talk about changing systems, we usually mean their change with respect to time.
End of digression.
Now, with all that said we don’t learn how the operator works here. I’ll fill you in on that when we get there at calc. For now, try to build up that intuition.
The other intuitive idea here is what dx/dt looks like on a graph. Imagine you have a sinusoid curve. So, a curve that wobbles up and down. Imagine the curve represents your position in the x direction over time. Since it’s sinuisoid, it bounces periodically from 1 to -1 over and over. Think of a man pacing back and forth, or a pendulum swinging, or a yoyo going up and down. In the motion of that pacing man, there are many different velocities. When he’s mid stride between two points, he’s going fairly fast. When he’s slowing to turn around, he briefly is going at a velocity of zero. The instantaneous velocity at a point is just how fast he’s going at that time.
Now here’s the cute part. Imagine you take two points in sinx and draw a line between them. The slope of that line represents the average velocity over that time period, right? The slope is the rise over the run, which in this case is the position change over the time change. Now, imagine you slowly shorten the distance between the two points you selected. Remind you of anything?
That’s right. We’re taking the limit. And when we take the limit, we get the instantaneous velocity, aka the derivative of the position! Once you zero in on a point, at that particular point you’ll have a certain value. Go ahead and try this at a few points on a graph yourself. You’ll quickly notice a pattern. At each point, you’ll have a line that runs just along the edge of the curve. That is, a line that is “tangent” to the curve. The slope of that line is the derivative at that point, aka the instantaneous velocity.
So, there are two big ideas here. First, the idea of the derivative, aka the instantaneous rate of change. Second, there’s the idea that the derivative of your position is your velocity. That is, the change of position with respect to time (that is, the d/dt of x) is the same as the velocity! So, practically speaking, if you have an equation that describes your position, you can use that equation to get the velocity. Neato!
Now, can you guys what would happen if you took dv/dt ?
BAM. CLIFFHANGER FOR TOMORROW!