## The math behind Michael Jordans legendary hang time Andy Peterson and Zack Patterson

Michael Jordan once said, quot;I don't know whether I'll fly or not. I know that when I'm in the air sometimes I feel like I don't everhave to come down.quot; But thanks to Isaac Newton, we know that what goes upmust eventually come down. In fact, the human limit on a flat surface for hang time, or the time from when your feet leavethe ground to when they touch down again,

is only about one second, and, yes, that even includes his airness, whose infamous dunk from the free throw line has been calculated at .92 seconds. And, of course, gravity is what's making itso hard to stay in the air longer. Earth's gravity pulls all nearby objectstowards the planet's surface, accelerating them at 9.8 meters per second squared. As soon as you jump,gravity is already pulling you back down.

Using what we know about gravity, we can derive a fairly simple equationthat models hang time. This equation states that the heightof a falling object above a surface is equal to the object's initial heightfrom the surface plus its initial velocity multiplied by how many secondsit's been in the air, plus half of the gravitational acceleration multiplied by the square of the numberof seconds spent in the air. Now we can use this equation to modelMJ's free throw dunk.

Say MJ starts, as one does,at zero meters off the ground, and jumps with an initial verticalvelocity of 4.51 meters per second. Let's see what happens if we modelthis equation on a coordinate grid. Since the formula is quadratic, the relationship between heightand time spent in the air has the shape of a parabola. So what does it tell us about MJ's dunké Well, the parabola's vertex shows ushis maximum height off the ground

at 1.038 meters, and the Xintercepts tell us when he took off and when he landed,with the difference being the hang time. It looks like Earth's gravitymakes it pretty hard for even MJ to get some solid hang time. But what if he were playing an away gamesomewhere else, somewhere faré Well, the gravitational accelerationon our nearest planetary neighbor, Venus, is 8.87 meters per second squared, pretty similar to Earth's.

If Michael jumped here with the sameforce as he did back on Earth, he would be able to get more than a meter off the ground, giving him a hang time of a little over one second. The competition on Jupiterwith its gravitational pull of 24.92 meters per second squaredwould be much less entertaining. Here, Michael wouldn't evenget a half meter off the ground, and would remain airbornea mere .41 seconds. But a game on the moonwould be quite spectacular.

## Vertical Asymptotes How mathbff

Hi guys! I'm Nancy. and I'm going to show you how to find the vertical asymptotes of a rational function. If you don't know what I mean by rational function just ignore that, you're probably in the right place. Actually all this stuff is a lot easier than it sounds. So don't worry. It's gonna be fine. Let me show you.

OK, so say you need to find the vertical asymptote or vertical asymptotes, if there's more than one. First of all. what is a vertical asymptoteé It's basically an invisible vertical line that your graph approaches and gets really really close to but never actually touches. Never quite reaches. So how do you find the vertical asymptoteé Well, if you have a rational function, the fraction form type,

which is the most common kind for this, you can use the same 3 steps always to find the vertical asymptote. OK. So here are the 3 steps you can always use to find the vertical asymptotes of a rational function. The first is to factor the top and bottom. If you can and all that you can. I'll show you what I mean.

The second is to cancel any common factors from the top and bottom to simplify. And the last step is to take the bottom, the denominator and set it equal to zero to find the vertical asymptotes. So let's try it for this function. The first step is to factor the numerator and denominator. if you can. And all that you can.

So you can't factor the top. in any way, but the bottom you can factor. It's a quadratic. And if you want more help on how to factor quadratics I have a tutorial for that. So you can check that out. But basically the idea is you want to break this into. 2 separate factors that start with x.

So x plus a number times x plus another number. OK, and what you want are 2 numbers here. 2 numbers here that. multiply to 6 and add to 5. And those 2 numbers would be 2 and 3. Since 2 times 3 is 6 and 2 plus 3 is 5. So in here we have 2 and 3. And that's all that you can factor for this rational function.

### The Jump Manual Review The Truth Everyone Should Know

Hey, what's goin' on guys!My name is Jesse, I'm 33 and I'm from Pensacola, Florida. I'm a basketball player from an early age. I'm here to share with you my testimonialabout the jump manual and how it helped me to increase my vertical. Before I discovered the jump manualand started using the jump manual workouts I was pretty much an average jumper. I could reach the backboardjumping off of One foot with Two hands and I could also touch the rimwhen jumping off of One foot with One hand

but still, I was nowhere close to dunking. and I started looking for waysto improve my vertical beacsue that just wasn't OK with me. I wanted to be a jumper that people wouldwant to gather around in the gym and watch him dunk. I wanted to be able to catch alleyoopsand throw the ball down like a hardcore proffesional basketball player. So I started doing some research becauseI just wasn't happy with where I was

and I knew that I had to do some researchto find a program or to find something that could really help me improvemy vertical so that I'll be able to dunk. â€˜cause this was my dream. and I did some research onlineand I found the jump manual. and I knew right away that this wasgoing to be the program for me. I knew that this gonna be the program thatwould take me to the next level. About the program. when I started the programimmediately I noticed gains in my vertical. and I didn't measure my vertical beforeI started using the jump manual

but I'm using the rim on the basketas a gauge to how high I'm jumping and how much I'm improving. and before I started using the jump manualI was barely reaching the rim on a good day jumping off of One foot. And then, as I started using the jump manualI gradually added inches and improving. my vertical has increased significantlyand I've been using the program for a few months nowand I can honestly say I'm a much better player now than what I waswhen I first started using the jump manual.

I can now jump off Two feetand dunk with One hand, I can also jump off Two feetand dunk with Two hands. I can stand underneath the basket flatfootedand jump up and dunk with both hands. and that's something that I never thought I could do. to be able to stand flatfooted andjump up and dunk with Two hands, that is just huge! Now I can jump out during a game with adefender on me and dunk the basketball, I'm also getting much more reboundsand I'm blocking a lot of shots as well,

and it's an amzaing feeling I gotta tell you. I feel much more confident andgenerally I'm just doing a lot of things now that I never thought I could do. and it's all thanks Jacob Hiller and his jump manual. it really helped me reach that level.and without the jump manual I wouldn't be able to doany of these things. I'm still improving though,my goal is to reach 45 inch vertical off of One foot,and 40 inches Off of Two feet.