Math Some Moar!
by, March 20th, 2010 at 07:12 PM (2234 Views)
Those who know me, know I have pet-peeves when it comes to Mathematics. The average fluency of mathematics should be much higher than it currently is. Better knowledge of basic logic, statistics, and problem solving skills reach out into your everyday life constantly. Forserious.
Here are three problems that I don't ever want to see again.
Problem 1: Action/Reaction Forces
Okay so I realize this plagues physics teachers everywhere... but Newton's 3rd Law is simple and elegant but very easy to be misused.
In a recent episode of Mythbusters, they were attempting the "shoot a gun out of someone's hand" myth. Being unsatisfied with using a dummy's hand... and being unsatisfied with hitting the gun with a baseball bat... they wanted to see what would happen with a bullet actually hitting the gun in a human's hand.
Since that was obviously unsafe, they said it would be entirely equivalent due to Newton's Third Law to fire a bullet from the gun (without the gun holder's knowledge of when) since the amount of force of a bullet hitting the gun is equal to the amount of force of a bullet leaving the gun.
That is untrue.
-The action/reaction force pair of a bullet striking a gun is the gun striking the bullet.
-The action/reaction force pair of a bullet being ejected by a gun is the gun ejecting the bullet. (yes the grammar is hard)
-The action/reaction force pair of a bullet striking a gun is not the bullet being ejected by the gun... the bullet isn't ejected by the gun... it's ejected by some other gun that's pointed at your gun.
This misunderstand was repeated by Jaime and the commentator for a good 5 minutes.
Easy/trivial way to prove this false: It ignores air resistance and friction as it travels through the air. That's negligible for most cases, but still.
Important reason why this is false:
Force is defined as The Change in Momentum Over Some Interval Of Time.
The momentum of the bullet is the same from when it leaves the barrel to when it hits its target... minus some friction and air resistance...
... so in order for the momentum and the force to be the same... the interval of time must be exactly the same.
This means that the bullet striking something and coming to a stop must occur over the same amount of time as the bullet took leaving the barrel of the first gun... and the force would need to be constant throughout the acceleration.
Sorry Mythbusters... I reject your physics and substitute my own.
Problem 2: Airplane on a Treadmill
An airplane on a treadmill will take off as long as the propeller's pulling velocity is greater than the friction of the wheels and axel (which is usually something like 0.02 for an airplane wheel) multiplied by the velocity of the treadmill by whatever velocity is required to make the airplane take off.
In other words:
V_prop + V_takeoff >/= u_kinetic * V_treadmill.
So say you need 10mph to take off... the treadmill spins at 10mph.
10mph*0.02 (coefficient of friction) = 0.2mph.
The prop needs to have a velocity of at least 10.2mph.
So if your plane can take off normally... you just need to increase its minimum velocity by ~2% to take off.
Protip: Your plane can put out more than 2% extra of the airspeed required for takeoff. So yes, the airplane will take off... deal with it. With perfectly frictionless and massless wheels, there would be no difference between the treadmill or not.
Why are people so stuck with this problem? People don't realize -- the speed at which the airplane travels isn't the speed at which the wheels turn. Your airplane is getting pulled forward by the prop pushing air backwards, which pulls the plane forwards... the wheels are getting spun by the ground speed.
If planes cared about the speed of the wheels on the ground... they would instantly crash to the ground as soon as they take off... right? The speed of the wheels is zero, right?
Problem 3: "Expert" Statistics
Learn to math.
"Violent kids play more violent videogames, so limit access to violent videogames to keep our children safe!"
No. Learn math.
Your statistics is based around the experiment of controlling the type of person you're surveying to be violent people. You then measure their access to violent videogames. You then claim that reducing access would reduce violent people.
Here's why math says that's an invalid statistic: You cannot vary a measurement and expect it to have the same correlation with the control variables. In fact, it's even possible to have a negative correlation when done in reverse.
The longer you leave your sprinkler on... the wetter your lawn gets. <-- Valid observation... forward correlation.
The wetter your lawn gets... the longer you left your sprinkler on. <-- Invalid logic... it could have rained.
The more it rained... the less you use your sprinkler. <-- Valid observeration, negative correlation.
Increase sprinkler -> Increase wetness
Increase wetness -> Decrease sprinkler (if it rained)
It is entirely possible that increasing access to violent media/videogames could decrease our violent crime rate. Proving that violent people play with violent media doesn't mean that violent media isn't influencing some of its users away from violent acts... it also doesn't mean it is... the statistic is simply meaningless because it's fundamentally malformed.
If you want to measure the influence access to violent media has on violent crime rates... you need to control access to violent media and measure violent crime rates. Simple. You cannot do it in reverse.
... oh and of course you need to do it for multiple sample spaces as well... just in case you pick the non-average case.
Quote an old physics prof of mine: It's easy to make a line of best fit for two points.
If there's anything you get out of this blog post... it's this:
Math works and helps you on a day to day basis if you use it and don't "invent" it.