Mathematically Diving

Diving can be done well or poorly, depending on how good the diver is. Sometimes, beginning divers will do "belly flops", smacking the water horizontally instead of cutting into it like a needle. Whether a dive is a good one usually depends on whether the diver went straight into the water or not. Experienced divers can do this without thinking much about it, as if it were like walking; beginners, however, have a lot more trouble.

In time, people get used to diving; machines, however, can't learn, and are always just as clumsy. If the machine contained a computer, it would need a computer program to help it dive. The program would need to use a math formula. Here's what we'll start out with: v₁= v₂ tan θ, where v₁ and v₂ represent forward and downward velocities, and θ is the vertical angle in degrees.

Here's how it works: the forward and downward velocities of a good dive have the same ratio as the sine and cosine of the vertical angle of the diver (represented by θ. See picture), so the formula is v₁ / v₂ = sin θ / cos θ. Since sin/cos = tan, the formula becomes v₁ / v₂ = tan θ; multiply both sides by v₂, and you get v₁ = v₂ tan θ.

To use the formula, you figure out what angle you'll be diving at and how fast you'll be falling when you hit the water (the downward velocity, v₂). When you put those numbers into the equation, you can do the math and get your forward velocity (v₁), so you know how fast you have to run off the diving board.

However, you might not know how fast you'll be falling (v₂) when you hit the water, so let's calculate it now. If you fell h feet in s seconds, your average speed in the time you fell was h ft per s sec, which is the same speed as h/s ft per sec. Since acceleration while falling is as good as a constant (ignoring air resistance), and you start at zero, your final speed is twice your average speed: 2h/s ft per sec. Insert that into the equation and you get:

Now we have another problem: s, the amount of time before you hit the water. To get rid of s, we just need to state it in terms of another variable. After falling for 1 second, your speed will be 32 ft per sec. After s seconds, your speed will be 32s ft per sec. We also know that your speed will be 2h/s ft per sec. Therefore, 2h/s = 32s. Work it out and you get s = √h / 4. The very complicated equation you see to the right is what we have now. And if we simplify it, we get the final equation you see below.

Comments

Dividing Paper Puzzle

When I was young, I would fold a sheet of letter paper in half, for origami projects. It occurred to me that the two halves looked almost the same as the whole sheet of paper - except they were smaller. I could see they weren't exactly the same shape; they were off by a little bit. But the idea stuck in my head. You can use a pen, instead of scissors, to halve the paper. Those rectangles all have the same shape, but are different sizes. One night when I was 12, I thought about my idea. I wondered if it was possible to have a sheet of paper that could be cut in half, resulting in 2 smaller versions of the same paper. That would be neat, to be able to cut a paper in half and get 2 papers that had the same exact shape. If that were possible, then you could cut those papers, too; and the resulting papers would have the same shape as all the other papers. You could keep cutting in half forever, and each paper, no matter how small, would have the same shape as all the ot...

Pluto No Longer on the Horizon

This morning, New Horizons became the first spacecraft to make a flyby observation of the Pluto system. During the mission, the spacecraft captured the most detailed photographs of Pluto's surface we've ever had, and possibly ever will have. It also found many new properties including size, mass, atmosphere, and surface composition. In a period of a few hours, we discovered more about Pluto than we've found in the 85 years since Clyde Tombaugh captured its first photograph. Before After (images credit: NASA) To complete this mission, the spacecraft flew for more than 9 years through the emptiness of space. This may sound like a long time, but it's actually amazingly quick. In fact, New Horizons set the record for the fastest speed at launch, and during the flyby, the spacecraft was moving at a rate of over 30,000 mph, or roughly 50 times the speed of sound. Picture an object twice as heavy as a grand piano moving 25 times faster than a bullet from a gun. Yikes. The man...

Gravity

Imagine the universe is filled with water. Instead of empty space, every inch of it contains pure water. No planets, no stars, only water. What happens? And what would happen if an air bubble formed? The answer to this question requires a basic understanding of gravity. Gravity is very important. It helps hold matter together, bends light, and distorts space-time (which, incidentally, is how it bends light). It also makes it possible to play football, and as Americans are big football fans, they would certainly agree that we couldn't live without it. Unfortunately, many Americans don't understand how gravity works. Admittedly, scientists haven't figured out a lot of things, but we do understand it well enough to make predictions and model physical events. One of the important things about gravity is that its strength is proportional to the inverse of the square of the distance. In other words, it gets weaker as you get farther away, based on the equation: F = c/d 2 where c ...

Cosmos Computed

Search This Blog