Applied Math Example - Definite Integral - Distance Traveled
This is my first of many articles on Applied Math. When I was starting off in college I was very frustrated learning all the math. It was all presented horribly, and I saw no application for it. For example when learning Integral Calculus, chances are huge that if you asked your professor what Integrals were for, they'd give you some technical words, then if you asked what it's actually FOR, then they'd say "to find the area under a curve" which is in direct agreement with the book you have. What the hell do I care about finding the crosshatched area under some arbitrary function printed off on some piece of paper? Well it turns out, quite a lot.
Unless you're a Math Major, or a teacher, or in some unlucky position, Integrals aren't about using the Substitution or Integration by Parts to find the area under the curve of problem #52. In Engineering, you use the Applied Math. You do what all that theory is actually FOR. The theory of Integrals has many applications.
Lets say an F1 car can brake at 4G at 325km/h, 3G at 225km/h, 2G at 125km/h, and 1G at 25km/h (or 202, 140, 78, and 16 miles for those of you that are from America and didn't suffer the agony of English Units in college). In case you were wondering, the braking ability of an F1 car goes down as speed goes down due to the downforce diminishing.
We want to know how much distance is traveled when going from 325km/h to a standstill.
When I was in highschool I would have just done 5 simple arithmatic problems and lumped them together (in 8th grade when I was in Algebra). If I wanted a more accurate number, and had more time on my hands, maybe I would have sliced it up in to around 10 chunks of finding the speed then, distance traveled during that time, and the new speed (later I made a program so I could make many slices). At the time I made up words like "very small numbers" and "local slopes" but years later I'd learn these methods were called Summations, Riemann Sums, and Integrals (plus other things I did were called Limits, Newton's Method, The Trapezoidal Rule, etc). Now with Excel or Matlab, it's easy to chop it in to thousands of chunks. All it takes is a little common sense, and trial and error.
Lets first tackle this problem with Summation and handle it with a Definate Integral next. We can start with a quick Khan video, since he explain it well enough.
I don't intend to teach you how to do math. I'm writing about Applied Math. This assumes you know the basic methods, or forgot them since they used to be so boring, or can learn them because you're smart and you have the internet.
Deriving all the necessary stuff for the summation process is more than I intend to do here in detail. For the Numerical Method, think for a while about how you would do it. When I was faced with this problem and saw the Differential Equation staring at me, I went straight to Excel to make a common sense guess. The method worked, and as usual, it was basically a Riemann Sum. The Excel file I made to estimate the distance traveled during the stop is attached below at the bottom of the article.
That is the end of the Numerical section. If this is still unclear to you and you're interested in knowing more, please let me know in the comment section below (User Registration is free, and helps avoid a ton of Spam comments).
So the location (x) of the car is a function of time. Makes sense. The velocity is the rate of change of location with respect to time. Acceleration is the rate of change of velocity with respect to time. This is all pretty straight forward. At the moment, all we know is the acceleration. A single G is 9.81m/s^2. Since at 4G that makes 39.24, 3G is 29.43, then 19.62 and 9.81. So in this case, acceleration is a function of velocity. Sure that's a bit more complicated but it shouldn't be a problem for what I'm trying to show you. Keep in mind, this now becomes a differential equation. I won't cover all the details of the differential equation, I'm just going to get the equation for velocity and we will have our "arbitrary equation" to "find the area underneath it". Using Wolfram Alpha I made a file you can download to see how to solve the Second Order Linear Differential Equation with Initial Values using Undetermined Coefficients. This solution to the Second Order Linear Differential Equation with Initial Values using Undetermined Coefficients is attached at the bottom of the article. If you aren't interested in that, here are the necessary equations.
We want to integrate the velocity equation, but first we need to find the range over which to integrate. We do that by finding the time at which the velocity is zero.
Now that we know the amount of time it takes to stop the F1 car, we can find the distance traveled by taking the Definate Integral of the Velocity Equation.
Below you can use a Widget I made on WolframAlpha to do the Definate Integral yourself.
Unfortunately the Widget runs slow when embedded, so you can check out the Widget on WolframAlpha Directly.
It can be seen that the results of the Integral from the Widget I made at Wolfram Alpha correlate with the results from the Excel Spreadsheet I made.
Differential Equation Solution
Multiple methods were used to find the distance traveled by an F1 car during heavy braking, and the data correlated. The solution was somewhat complex since I assumed the acceleration was changing with velocity due to diminishing downforce. I did my best to leave the most complicated stuff out of the way of beginners, but where readers more familiar with Differential Equations could find it. I made a pretty cool widget and a couple downloads for you guys.
I hope you enjoyed this and it made sense and helped you. Please let me know it did. I'd appreciate the feedback because I do this to help (it's hard to find this sort of stuff online) and without your feedback I'll assume this isn't helping and I'll move on to other things :)
You can find the list of all my Applied Math Examples List - Methods - Real World Uses.
The next article in this series has been added. Click the image above to continue to my data on Real F1 Telemetry Data.