**Test
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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.

- 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.
- 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?
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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

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