I show how a Definite Integral can be used to find the distance traveled by an F1 car under Braking.
I wanted to make an improvement on my previous article about finding the distance it takes an F1 car to stop.
This time, I managed to get my hands on some F1 Telemetry Data. I was pointed to some Telemetry Data from Caterham F1 which I used to improve on the previous assumption that Acceleration varied in a more simple manner as Velocity decreased. I will write a more in depth article later on how I transformed the data from the image below in to what we will be doing here.
For this Applied Math Example I took direct F1 Telemetry Data in order to better model the braking. I knew the linear braking model was incorrect since Lift varies with the square of Velocity, but since I didn't have accurate data I didn't see the need to push hard for a real good prediction. I only wanted a simple equation for you guys to Integrate. However this time, with F1 Telemetry Data Analysis, I got a more direct polynomial that I was able to Integrate before checking it against the real F1 performance. The purpose of this Applied Math Example is to show there are many interesting real life uses for Math, and that Math isn't nearly as pointless and difficult as you were led to believe by those teaching you at school.
Using the F1 Telemetry Data, generate an Aero Model and use it to find the distance it takes to slow an F1 Race Car from 88m/s to 20.65m/s. Check to see that the end result is reasonably close with the real life F1 Telemetry Data.
Just a reminder - my goal is not to teach you how to do Math. There are others that can do that better than me. The article linked above has a good training video. If you want to learn how to do Math, you can find help, if you want to learn to Apply it, I think you might have come to the right place.
The "Area Under a Curve" can mean lots of things. In this case, the area under the Velocity curve represents the total Distance Traveled. Think about it a bit, you are multiplying (m/s)*s. I will give you Velocity as a Function of Time, and you'll use that wretched thing called an Integral to actually do something with real world application. For an extra challenge, try getting your own Velocity Function. I used the big braking zone to right of the image below.
Caterham F1 Telemetry Data - Spa
For a higher resolution version of the image above, go to the Download at the bottom of the article.
I won't go in to detail, but the F1 Telemetry Data were presented as a function of position rather than time. Due to this I had to use some methods to convert everything to be a function of time, starting by extracting the data through a Computer Vision Algorithm I wrote. After doing all the back end work, which I will describe in a later article, I got the equation we will be using for velocity. First, you need to solve for the correct final time. The F1 Telemetry Data reached the end of the braking zone at 20.65m/s. Use that to find the time domain over which to Integrate, and you should end up with 153.7m. For a reference, I got from the F1 Telemetry Data that the distance was 153.6m. It sort of depends on how you deal with the data due to the "low" resolution of the image, but no matter how you slice it what we got here was within 5%. Not bad!
If you need a calculator, you can use the Integral Widget I made at Wolfram Alpha.
So when you are looking at problem #85 on page 635, remember that the stupid f(x) garbage you are looking at could be representing something very interesting, such as an approximate function representing F1 Telemetry Data. I wish I would have known that when I was taking Calculus! Suddenly the area under a curve becomes using a function of velocity with respect to time and Integrate it in order to find the distance travelled. Math isn't as boring and pointless as you were taught, Integrals, and even more complicated things, have real life uses.
For more Applied Math Examples:
As always, I do this to help. Please give me feedback through Comments, Likes, Shares etc so I can see that this is helpful, otherwise I'll have to find a more useful way to help people. Having this fall on deaf ears won't do any of us any good, so let me know if I'm accomplishing my goal of giving you something enjoyable and educational. If it isn't accomplishing my goal, please let me know what I can do that will .