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Space Jump

Buddhas Hand

By Buddhas Hand
in Off-Topic General

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Canuckerbird Abbotsford Prospect

Canuckerbird

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Judging from the earlier post explaining in detail all the horrible things that could go wrong, it sounds like there's a good chance we'll see a disaster if and when he actually jumps. Hopefully everything works out and we push human limitations even further.

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Buddhas Hand

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Stratos Space Jump: Can You Fall Faster Than the Speed of Sound?

It seems like the Red Bull Stratos jump (redbullstratos.com) is back on track. For those of you not familiar, the basic idea is fairly simple. Felix Baumgartner will take a balloon ride up to an altitude of 120,000 feet and then jump out. The project was temporarily on hold due to some legal issues, but it seems to be settled now.

Yes, I know I have already talked about the Stratos jump many times. But that was then, this is now. How about I try to answer some of the questions that might come up?

Will he jump from outer space?

I guess the answer to this question depends on how you define ‘outer space.’ Perhaps the common definition is “the region where there is no atmosphere.” The problem with this is that the Earth’s atmosphere doesn’t just abruptly end. The transition from space to the Earth’s atmosphere is more like a hill rather than a step.

If you want some values, most people consider the International Space Station to be in outer space. This orbits the Earth at about 300 km above the surface (or 980,000 feet). You could say all the way down to 100 km (330,000 feet) is pretty close to space. So, 120,000 feet isn’t quite there. Don’t get me wrong. It’s still way up there.

But what about the density of air that high? On the surface, air has a density around 1.2 kg/m3. At 120,000 feet the density is only 7.3 x 10-4 kg/m3. Since everyone always loves a graph, here is a plot of the density as a function of height (based on this density model).

air_den_feetpng.jpg

So, if you wanted to call the jumping height “space” it wouldn’t be such a terrible thing. You would definitely need a space suit that high, so it could be space. Right?

How do you get to that altitude?

There are some options. The first and obvious option is a rocket. Why not some type of airplane? Well, most airplanes need this thing called “air” to work. As you can see from the previous question, there isn’t much air up there. So, other than a rocket, the best choice is a balloon. But wait. Didn’t I just say there isn’t much air? Yes, I did. Balloons also need air, but if you get a big enough balloon you can make it work.

If you think about the balloon at some particular height, the forces on it would be something like this.

2010_05_26_untitled.jpeg

There is the gravitational force and a force we like to call the buoyancy force. Really, the buoyancy force is the result of the air colliding more with the bottom of the balloon than it does with the top. The bigger the balloon, the more collisions and the greater the buoyancy force. However, there is a problem. If you just take a balloon and blow it up with air, the gravitational force on the balloon also increases with the size of the balloon. The trick is to use a gas with a lower density than air. In this case, that gas is helium.

Still, as you can see above, the density of air at 120,000 feet is really low. With a low density, there not as many collisions between the air and the balloon. The result is you need a bigger balloon (which unfortunately has more mass). So, in the end, you need a balloon about 80 meters (over 250 feet) across when fully inflated to lift a jumper and life support capsule to that height.

How much less will the gravity be at 120,000 feet?

Ok, so the air is pretty thin up there, but what about gravity? Clearly there IS gravity in space. This is the force that causes the moon to orbit the Earth and the Earth to orbit the Sun. On a side note, it is fairly common for people to think there ISN’T any gravity in space.

Anyway, the gravitational force depends on the distance between the objects (for spherical objects at least). If you double the distance between the center of a planet and a spaceship, the gravitational force will only be one fourth as much. The key here is “center of the planet”. So, if I am 10 feet above the surface of the Earth and I double this height to 20 feet, how far did I move from the center of the Earth? The answer: not at all (to first approximation). The reason? The Earth is huge. It has a radius of about 6.38 x 106 meters (or 2.09 x 107 feet).

Go ahead and try this. Put 2.09 x 107 + 100 in your calculator. What do you get? You get 2.09 x 107. This is because your calculator rounds off the actual value.

Fine. The gravitational force doesn’t change too much near the surface of the Earth. But what about 120,000 feet? Well, a 1 kg mass has a gravitational force of about 9.8 Newtons (2.2 pounds) on the surface of the Earth. At an altitude of 120,000 feet the gravitational force would be 9.68 Newtons (2.18 pounds). This 98.8% the value at the surface. So the answer is that the gravitational force at 120,000 feet is pretty much the same as on Earth.

Oh, maybe I should add that the gravitational force on astronauts in the International Space Station is about 91% the value at the surface. Then why do astronauts float around in space? I am glad you asked (your answer).

What is the speed of sound?

One of the cool things about the Red Bull Stratos jump is that it will be a chance for a falling human to fall faster than the speed of sound. So, what is the speed of sound? I guess you could ask “what is sound?” – but maybe I will look at that later.

If you think about introductory physics, the speed of sound is often stated as being around 340 m/s or 760 mph. This is the value for the speed of sound at normal temperatures and pressures (like near the surface of the Earth). But sound is an interaction between air molecules – so it really depends on what they are doing (and it really isn’t so simple). However, there is one model for the speed of sound that says it is proportional to the temperature (this is just a model – but it works fairly well).

The higher you go, the lower the temperature (up to a point). Using the same model for the density of air, I can get the temperature and thus the speed of sound. Here is a plot of speed of sound as a function of altitude.

speedsoundpng.jpg

At 120,000 feet the speed of sound is only around 200 m/s (450 mph).

Can he fall faster than the speed of sound?

Here is the real question that you have been waiting for. The answer is yes (probably). To understand how, lets look at the forces on Felix right after he leaves the balloon.

drawingskey_1.jpg

Since he isn’t really moving (yet) and there isn’t much air anyway, there is just the gravitational force on him. Since this force is down, it causes him to start moving faster and faster as he travels down.

As he starting going faster, there is an air resistance force. You have probably felt this force when you put your hand out of a moving car window. The faster you go, the faster the force. However it also depends on the density of air. So, at the begging of the jump, the forces might look like this:

drawingskey_2.jpg

Since there is an air resistance force in the opposite direction to the gravitational force, it essentially makes the total force smaller (but still down). This means that he will still speed up, but the rate that he speeds up will be less. The key point is that he is STILL speeding up and getting faster. Oh, this is the part that he could go faster than the speed of sound.

He can’t keep speed up forever. Eventually, his speed will get large and the density of air will increase as he gets lower. At some point the air resistance force becomes larger than the gravitational force like this.

drawingskey_3.jpg

Now that the force is in the opposite direction as his speed, he slows down. Eventually, he will slow down to the point where the air resistance force and the gravitational force are the same. At this point, he will not speed up nor will he slow down. This is called terminal velocity.

I am aware that I haven’t answered the question. What about the speed of sound? Honestly, finding the speed isn’t quite that easy. Really, the best way to do a problem like this is to break it into a whole bunch of small steps and let a computer do the work.

Doing, that here is a plot of Felix’s speed as a function of time. I have included a curve showing the speed of sound for the altitude he is at during that time.

ull_seminarkey.jpg

According to my calculation (which does have some assumptions in there) he will be going faster than the local speed of sound for about 1 minute. He will also go faster than the speed of sound at the surface at one point.

But don’t sky divers fall around 120 mph?

Sky divers? Yes. Felix? No. Why? Because as he falls, the density of air (and thus the air resistance) changes. He doesn’t stay at the same air density long enough to slow down to a terminal velocity for the first part of the jump. Of course, eventually he will. Here is a plot of his speed along with the local terminal speed.

2010_02_23_untitled_7.jpeg

You might notice that I cut off the first part of the graph. This was because the terminal speed at this high altitude was ridiculously large. It made the graph look silly

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