The Mathematical Breakdown of Road Rage
You know, I hated math, right? I can’t do it, I just can’t. It’s never going to be used for anything. When am I going to use geometry again? Thanks for teaching me about triangles!
I hear this anytime I bring up what I studied. I never bring it up anymore. No good can come of this.
The answer is no and your explanation is more absurd than you think. You don’t hate math. You’d literally be brainless without math. The only reason you can understand this concoction of words I’m spewing on this white or black page is because mathematics taught you about order and patterns and symmetry.
Syntax. Think about it. Can you count the number of letters in each of the words? Yes, you can. Look at that, you’re doing math. Seems like you’re okay at it. Can you count the number of words, too? Holy moly, look at that!
It’s swirling around you at all times. If you decide to cut someone off on the freeway, you’re going through the trouble of mathematics without the formal environment of symbols.
Let’s say you find yourself on the freeway.
Driving along, in the fast lane.
All of a sudden, some jerk cuts in front of you and slows you down.
You honk and you honk, this person won’t go above 50 miles per hour.
The speed limit is 70. Everybody else is passing you two.
You and I and everybody else already know what to do.
You are going to swerve out of the lane, speed up, and then overtake this jerk.
But how do you know how to do that?
How do you know how to overtake someone?
You’ll push the pedal to increase the speed, but not so much so that your new speed is dangerous.
How long will that take you?
Approximately when can you take your lane back?
What if there is a car slightly above you guys in the lane to the right?
You’re going to use calculus without knowing it.
If you took calculus in high school or in early college, you’ll recognize
This is just a derivative.
In high school, they teach you that the analog of this is speed.
It is the rate at which the position function is changing at the point, x, in time.
We will ignore the position function for simplicity’s sake, but let’s say, for those who remember, that the symbol above represents speed, the derivative of your position function.
Now you want to make sure you’re passing this jerk successfully without stressing yourself out any further.
Your speed can be modeled as
The jerk’s speed can be modeled as
You are both going at the same speed, so this is the same as saying those two speeds are equal
But we also know about the derivative of the derivative.
We know that the analog for this is acceleration.
We know that this is the rate of change in our rate of change. In terms of passing this jerk, we know it is the change of our speed.
In other words, we know that
Where we can recognize the second derivative to be the acceleration of the car we are in at point x for some point in time.
Specifically, we need every point in time after we move out of the current lane to be faster than the jerk in the original lane.
We need to have a higher speed.
In other words, we need
You could even go a step further. You could easily say that since the jerk has been going 50 miles per hour the entire time, his or her acceleration has been 0.
Why do we know this? Because, for acceleration to be positive, you would need the rate of your speed to increase. From the situation we are in, we know that is not true.
We know from observation that the maximum value for the jerk’s acceleration is
Why do we know this? Because we know that since the jerk’s speed has been the same throughout the entire time we’ve been angry. This tells us that his or her acceleration is at most zero, and, at worst, going even slower than previously, which would tell us that his or her acceleration is negative (decelerating).
We can also safely say that the jerk will continue this pace of constant zero acceleration (or deceleration) indefinitely — judging from how angry they’ve made us that we’ve had to overtake them in the first place and create this post about them.
So all we’ll need is
And, depending on how positive this value is, and for how long you accelerate, you reach and overtake the original car on your left that you wanted to pass.
You could get exact with the original position function to model and assume precisely when you’ll be able to safely overtake this person, at what time, and at what speed to accelerate to.
You could even throw in obstacles with cars obstructing you from accelerating passed the original slow driver.
We use models like these every single day without actually going through the math. Whether it is when you’re buying food, or weighing the odds of a decision, or partitioning the amount of time you want to spend doing certain things.
You’re using the laws of mathematics to help govern those choices in the background.
It just doesn’t always require you to write it down first. It doesn’t even require you to formally recognize it.
You can’t hate math. You use it every day, even if you tell yourself it’s not useful or practical. You need it to exist. If you get angry, you’ll probably prove this point.
It’s the basis for the foundation of the sidewalk you walk on, the roads you drive your car on, the machinery that goes on under the hood of your car to propel you forward, the integrity of the home you sleep in. Everything in our life rests on the principles of mathematics, from the material world to the material in our minds.
Your relationship with mathematics is far more intimate than you think. It has roots so deep in our everyday life that it’s invisible.
The greatest irony is to use math to proclaim its very own uselessness.