Test

1 03 2010

Test

 

 

Today, I’m going to introduce a new segment where I prove or disprove questions using math. I mean seriously, what can be more fun than taking things you enjoy and applying math to them? I personally can’t think of anything – I feel like a friggin’ Myth Buster!

NOTE: If you are one of the people who still choose to believe that you live in the delusional world where math does not seep into every facet of your life, and want to continue to hold on to your hopes and dreams as if they were possibly conceivable, you have two choices.

  1. Leave. Walk away. Continue to live in your little world where you believe anything is possible. Let’s just see how far that takes you.
  2. Swallow your pride and embrace math’s role in everyday life. However, if you’re still not fully convinced, prepare yourself with another delightful proof, like this one disproving the existence of Santa.

Now that I have broken you down completely and crushed any (if not all) of your hopes and dreams, let’s continue.

For anyone that has ever played baseball as a little kid, you always wondered (if even only in your head) how far you could hit the ball. Could you hit a little-league home run (roughly 200 feet)? Could you hit a major league home run (roughly twice that)? Could you hit the ball into space? Now hitting a baseball into space seems impossible. But at that age didn’t all three of them sound impossible? Yet, from experience, you know that the first two can be done. But do we really know that the third one can’t be done? You may have tried yourself, but that just proves that you couldn’t do it, not that it was impossible. So let’s start, can you hit a baseball into outer space?

Before we prove this, we will have to find out how hard one would need to hit the ball to hit it out of a major league park, so we have a comparison. But let’s start with a simple example and we’ll build up from there. (If you know some calculus or physics, you can skip to here).

When you throw a ball up in the air, what does it do? Well, it comes back down. And why is that class? If you said gravity, then we are on our way to solving this. Gravity is all around us and understanding the basics of it, is essential to this proof. So let’s take some time getting to learn gravity – think speed date, not dinner and a movie. Gravity is one of the four fundamental forces in our universe and has the same effect on every particle, say an atom or a baseball. You may remember a teacher or an old School House Rock episode saying that if dropped a bowling ball and a feather from equal heights, they’d hit the ground at the same time, even though they weigh differently. However, if you ever tried to mimic these results, you’d probably fail due to existence of air being in the way (air resistance). This can be seen be dropping a piece of paper – if you drop it one way, it slowly glides to floor and if you drop it length-wise it falls straight down. Now let’s go back to our bowling ball experiment, if there was no air (also known as a vacuum) in the environment (allowing for the slow gliding of the feather), then both objects would hit the ground at the same time. If you still don’t believe me, watch this video of a classroom gravity experiment (they take all the air out of a plastic tube). As I mentioned earlier, gravity effects ever particle the same way – it is a constant force pulling everything down. And this constant force has been measured to be 9.80665 meters per second per second (for all intents and purposes gravity will be a downward acceleration of or ) so the bowling ball and the feather are both pulled to the Earth at the same rate, thus hitting the ground at the same time. Don’t worry about the number for now or the term “acceleration”, we’ll go more into that in a bit.

So let’s take a step back again and look back at the throwing the ball in the air example. When you throw it straight up, it comes straight down. However, if you throw the ball to a friend, it goes up and then comes down in a parabolic arc.

This would be the case in any scenario of a ball moving at some angle up off the ground. So we can apply this diagram to hitting a baseball as well.


Now if you threw/hit the ball at a low angle, common sense would say it wouldn’t go that far because it would hit the ground first, right? And if you threw/hit it too high (say a pop-fly), the ball would get some height but not that much distance. So at which angle would one hit the ball for optimum distance? For that, we are going to need some definitions and formulas.

Our first term is going to be velocity. Concisely, velocity is the same as speed with the exception that velocity also denotes direction. For example, if you are going a speed of 50mph in your car you could also say your car is going a velocity of 50mph. Now let’s say as you are driving, a car on the other side of the road drives past you going a speed of 50mph but in the opposite direction. Then you could say that that car has a velocity of -50mph because it is going in the opposite direction. The positive/negative signs on velocity decide which direction the object in question is traveling. Speed, on the other hand, is always positive that’s why your car’s speedometer starts at 0 mph.

While we are on the subject, let’s take some time breaking apart mph. Mph obviously stands for miles per hour, but what exactly does that mean? Well if you were going 5 mph for a whole hour, you’d be 5 miles from where you started, right? So it’s understandable to say that mph is the rate in which one’s position moves over a given time or the rate of change of one’s position. In math, “rate of change” is a phrase that normally denotes a derivative and that’s exactly what this is – mph is the derivative of miles or more formally, velocity is the derivative of position. So basically, velocity is how fast one’s position changes.

More notably in math notation, it would look like this:

Now if we had the velocity and wanted to know the change in position we can take an anti-derivative or an integral.

Congratulations, you just did calculus! Don’t worry if all the extra terms and symbols don’t make sense, all that matters is that you understand the concept and the relationships between velocity and position.

Before we go any further we are going to need, we are going to need one more relationship, acceleration. Acceleration is the derivative of velocity, that is how fast one’s velocity changes. So using from what we learned above we can write:

And from the rules above we now know that:

And as I mentioned earlier, gravity is the downward acceleration of or in shorthand, gravity because the negative denotes direction.

NOTE: “meter” is a metric unit and is more commonly used in math and physics, over such units like “feet” or miles”. For the remainder of the calculations I will use the metric system as it flows nicer, but I will take care to provide the results, in terms more common to America’s unit system.

So we stated this relationship between position, velocity and acceleration, but they don’t look anything like equations you did back in grade school. So we are going to have to expand on the formulas above. First, let’s start with velocity, and by expanding further we get:

We can do the same to solve for position.

We now have our two equations but because we are working in two dimensions (the left-right direction and the up-down direction) we are going to have four equations. First, we’ll label each direction, the left-right direction will be the x-direction and the up-down direction will the y direction.

Then we’ll set up our scenario in picture form; where is velocity, is the height off the ground of the hit, is the height of the centerfield wall and is the distance from the hit to the wall.


 

 

 

 

Finally, we’ll separate our equations into the x-direction and y-direction.

But since the ball is not accelerating or decelerating in the x-direction, . In addition, if we set our initial position to 0, that makes our x-equations:

And since we know that gravity is accelerating at a rate of and our initial y value is , our new y-equations become:

Where

 





Conquistador (Pt. Two)

15 02 2010

Now for all you non-garden lovers out there – I didn’t forget about you!

There are other ways to use conquistador, and not all of them are limited to gardens and triangles. For instance, when you go to the store and see a bottle of soda and a pack of can sodas – do you debate before you buy them? Well maybe you should! Waldbaums sells 2-liter bottles of Coke for $1.89 a piece, yet it sells its 12-pack of cans (each with 12 fluid ounces) for $5.99. Now with a couple simple calculations:

Doing the same for the 12-pack we see that:

I know, I know – all that work for just a one-cent difference. Plus, I don’t expect you to go the supermarket with a calculator or pen and paper, so why bother with the calculation? Well let’s see, if you can buy two bottles of Coke for $3.78 ($2.21 less than the 12-pack) and end up with 135.256 fluid ounces, which is roughly 11 and a quarter cans. So in a way, by buying the 12-pack you are paying $2.21 for the extra ¾ of a can! That may just be chump change to you, but it can add up.

Let’s assume you go through two bottles of soda every week:


So for taking 10 minutes to sit down and work a simple equation out, you basically made $114.92! Move over Richie Rich!.*

*Actually Richie Rich will not have to “move over” because at $114.92 a year, it would take you a minimum of 8,701,705,535 years to surpass his wealth.

So for taking 10 minutes to sit down and figure that out – you basically made $114.92 – that’s equivalent to making $689.76 an hour!* Move over Richie Rich.**





Conquistador (Pt. One)

4 02 2010

Math can be a very scary word. Just the sight of it can cause people to fall asleep, tune out or just run away with their tails between their legs. However, you use math everyday from counting calories to judging if you have enough time to beat the light. Numbers and odds are constantly running through your head and it has become such a second nature that you probably don’t even realize it. So if you use math every day, why do you still cringe at the word? Maybe, it’s not the subject that’s scary- maybe it’s the word “math.” Let’s shake things up a bit and give your brow a rest (it’s been worrying for the last hundred words or so) by substituting another word for “math” the rest of this post. According to Wordcount.com (a collection of 86,800 of the most frequently used words in the English language), “conquistador” is the least frequently used word in the English language, and I see no better way to boost its rank then by letting it take math’s place (#46,914) for a while. For those math friendly-minds out there, math=conquistador.


So now that that’s cleared up, on to business. This blog is about how we all use conquistador everyday and how we should embrace it because it only makes life easier. You can do some pretty interesting things with conquistador, if you give it a chance. Remember when you were learning triangles in grade school and you asked the teacher when you’d ever going to use this in life? You weren’t gonna be a trianglologist or get your PhD in triangle studies when you grew up – it just seemed useless. But let’s say you have limited space in your backyard and were planning on building a triangle-shaped garden back there. You plan on having it be six feet long by eight feet wide, but you don’t know how long the last side has to be to complete your triangle. Now you could go through the trouble of laying both sides out and then measuring, or even buying an abundance of material for the last side and hoping it matches up. On the other hand, if you have a second, you can apply the Pythagorean theorem. The Pythagorean theorem states that if you square the legs of a triangle and add them you get the square of the longer side (the hypotenuse).


Now in our case we have:

And because

Therefore, our last side must be ten feet long to complete our garden. Easier than moving six and eight foot barriers, if you ask me. Trust me this is just the tip of the iceberg, conquistador gets much more interesting!





More to Come

28 01 2010

You hate math, don’t pretend you don’t. But what you don’t know, is how often you should use math everyday. With the help of this blog, you will learn how fun math can be and how essential it is to everyday life.