Dot Physics

Stratos Space Jump

Red Bull is sponsoring this sky dive from really really really high up – Stratos: Mission to the Edge of Space. Seems dangerous. The basic idea is that Felix Baumgartner will take a balloon ride up to 120,000 feet and jump out. Here are some questions:

  • Will he reach supersonic speeds?
  • The Red Bull site says: “can Felix react to a 35 second acceleration to mach 1?”
  • How about the claim that he will free fall for 5 minutes and 35 seconds? That seems pretty short.
  • In 1960, Joe Kittinger jumped from 102,800 feet. Will 20,000 feet make a large difference?

Assumptions

Clearly, this can be a difficult problem. I will start with the following assumptions (which clearly you could argue may or may not be valid).

Model for air resistance: I will use the following for the magnitude of the air resistance force:

i-d7d39adb0a4fb7df70381ce1c496c258-2010-02-19_la_te_xi_t_1_9.jpg

Why can this be a problem? If you throw a ball, or shoot a bullet at subsonic speeds, this model probably works just fine. However, what if you are in a low density gas and traveling faster than the speed of sound? Maybe this isn’t the best model. I am going to use it anyway. Perhaps it won’t give the most accurate results, but it gives me something to start with.

Constant A and C: In the air resistance model above, A is the cross sectional area and C is some coefficient that depends on the shape of the object. I am going to assume these are the same for a normal sky diver. I will also assume that they are constant during the fall. Why would they change? Well, if the jumper goes into a dive or flattens out, then C would change. A and or C would also change if the jumper release a small stabilizing chute (which I am pretty sure the 1960′s guy did).

Function for Density and Temperature of Air and Speed of Sound: Here is the problem. At 120,000 feet, the density of air is not the same as at sea level. Wikipedia has an explanation of my density as a function of altitude model that I will use. I will also use this to model the speed of sound as a function of altitude. This site and Wikipedia suggest the following model for the speed of sound in air (which is just temperature dependent):

i-760971fb61351db8cae3e0535db15e5e-2010-02-19_la_te_xi_t_1_10.jpg

Maybe that is not the best model, but I am sticking with it.

Method

So, here my plan. I will use python to run a numerical calculation (here is an review of numerical calculations). Here are the basic steps:

  • Calculate the air density and temperature
  • Calculate the net force (air resistance plus gravity – real gravity)
  • Calculate the new velocity
  • Calculate the new position
  • Update stuff (like time)
  • Repeat

I hinted above, but just to be clear, I am using the more realistic expression for gravity:

i-34b19c3890f0bb4c4a3a1f868d033879-2010-02-19_la_te_xi_t_1_11.jpg

Now I will address the questions from above.

Acceleration to mach 1

First, does he reach mach 1? Here is a plot of his speed vs. time along with the speed of sound for that time (at that altitude).

i-668be77dd02817b25cd6f5d8e1765b71-2010-02-19_untitled_1.jpg

At a time of 21 seconds, the jumper would reach a speed of 205 m/s, and this is the speed of sound at that altitude. So, this is different than what Red Bull claims as a time of 35 seconds. But maybe they are talking about how long it would take to get to a speed of sound at sea level of about 340 m/s. So, according to my numerical calculation, the jumper would reach a speed of 340 m/s in about 38 seconds. Perhaps that is what they were thinking about.

Then what is his acceleration? Let me first calculate the average acceleration (in the y-direction):

i-ef8162f85c66ea16ddde26909784eefd-2010-02-19_la_te_xi_t_1_12.jpg

And if I use the speed of sound at sea level:

i-fc14687b3707d5acfebaa9305f1bf11f-2010-02-19_la_te_xi_t_1_13.jpg

Both of these accelerations seem reasonable. First, they are at a very high altitude, so there is not too much air resistance. Felix should be able to handle this acceleration just fine. It is not very large and also he wouldn’t even feel the gravitational force since it pulls the same on all parts of his body. Here is a post on the difference between weight and apparent weight. Ok. Enough about this question.

What about free fall time?

This is pretty straight forward – I just need to change the program to plot height and time.

i-a4ffaeb1c6a20cd49fc083eee43c7592-2010-02-19_redbullheightpng.jpg

From this, it will take around 200 seconds to get to 5,500 meters – this is about 3 minutes and 19 seconds. Not quite the same time they predict. Joe (in 1960) jumped from a lower altitude and took over 4 minutes. Something must be not quite right. It could be my A*C term, or it could be my density of air function. Overall a free fall time of over 5 minutes seems reasonable.

Will 20,000 feet make a big difference?

Here is a plot of the speed of a jumper from 120,000 feet and from 102,000 feet as a function of height.

i-90eb5720a840ece04bee8386feb560fb-2010-02-19_compare_speed_distancepng.jpg

So, it looks like starting 20,000 feet higher can indeed make a big difference. This is because the jumper will have a lot more time up high to gain speed where the air resistance will not be as significant. (the green line is the speed of sound vs. altitude).

I don’t think my model is too bad. From the Joe-1960 jump, he reached a maximum speed of 614 mph (274 m/s). From the plot, this is about the max speed for the red curve. I am happy enough.

PS: I don’t know when Felix is jumping – for all I know he has already jumped. Hopefully, his mission will be successful.

Comments

  1. #1 sparc
    February 19, 2010

    Joe (in 1960) jumped from a lower altitude and took over 4 minutes. Something must be not quite right. It could be my A*C term, or it could be my density of air function.

    To my best knowledge Kittinger used a small stabilizing parachute that may have decreased the velocity of his then not so free fall.

  2. #2 William
    February 21, 2010

    Nice post, Thanks

  3. #3 Jesse P.
    February 23, 2010

    Since the acceleration is down, and nearly g, He would actually experience near weightlessness. The most important ‘damaging force’ will not be a uniform wind but shear (different speeds at different parts of the body). In addition to tugging apart forces, wind shear can put you into a spin (death spiral) may be very difficult to get out of – you could black out from rotational effects, e.g.
    He will generate a Mach cone too- I don’t know what the effects would be at that altitude, and how much differential force/torque might be generated, though the fact that the acceleration is near g suggests that it won’t be too much.

  4. #4 me
    February 24, 2010

    looks like someone drank way too many red bulls could not sleep and didnt have anything else to do but try and belittle a huge feat that will help out future space explorations and understanding. The stuff you wrote, I am NO rocket scientist so the average joe just did what I did; read the first part in english where you question the project and then skipped to the bottom. Good for you why dont you get in touch with all the experts and tell them they can’t do it. I am sure they will listen to you. LOL

  5. #5 Steve Street
    February 24, 2010

    I remember as a small boy reading about Joe Kittinger. In fact Joe broke many records, including high altitude jumping.
    My concern for Felix is as stated above, the dreaded spin. I wonder what kind of attitude and directional control of his body will he be employing?
    Both men have brass body parts. Salute to them both!

  6. #6 xx360xx
    March 7, 2010

    hi
    just some information:
    The speed achieved by human body in free fall is the function of two factors, body
    weight and body orientation. In stable, belly to earth position, terminal velocity of the
    human body is about 120 mph. Stable freefly head down position has a terminal speed
    around 150-180 mph. Further minimizing body drag and streamlining the body position
    allows to reach greater speeds in vicinity of 300 mph
    Word record is more than 500km/h with an exit altitude of 14000ft.
    A skydiver (in a fixed freefall position) who has a terminal fallrate of
    62 meters/sec at 10,000 feet. will have a terminal fallrate of 50 meters/sec at
    3,000 feet.
    there is a lots of variable for a jump at this altitude!
    i’m a skydiver and i’m sure felix can reach supersonic speed without any problem!
    5 minutes and 35 seconds it’s not short when all skydiver can easily reach 300km/h in 20s!!
    and yes 20,000 feet make a large difference!
    try skidiving i’m sure you will understand more about this project!

  7. #7 zr1
    March 16, 2010

    Just a note on the speed of sound.

    Enjoyed your calculations, but I believe your numbers for the speed of sound at altitude are not correct as it is nearer 300 m/sec at those altitudes instead of the 205 m/sec mentioned above. The speed of sound does not have a direct relationship with altitude. The speed of sound goes down as one goes higher in altitude until about 10.7k meters where it is 295 m/sec, then the speed of sound rises after that. Kittinger’s max velocity on his jump in 1959 was 274 m/sec. but was subsonic at Mach 0.93. But I’m still hopeful that the terminal velocity for this jump goes transonic.

    If this list shows correctly (and converting to meters):

    Altitude (m) – speed of sound (m/sec)
    36.5k – 311
    33.5k – 305
    31.1k – 302 (Kittinger’s jump)
    27.4k – 300 (RBD’s est. max velocity at 340 m/sec)
    24.4k – 298
    21.3k – 296
    18.3k – 295
    15.2k – 295
    12.2k – 295
    9.1k – 303
    6.1k – 316
    3.0k – 328
    0 – 340

    Here’s an online calculator to estimate atmospheric properties:
    http://www.aerospaceweb.org/design/scripts/atmosphere/

  8. #8 Long drop
    May 3, 2010

    Will he heat up? A 100kg man at 35,000 metres has about 35 MJ of gravitational potential energy, i.e enough energy to boil 100 kg of water. To lose this much energy in 335 seconds of fall is about 100kW. So does his suit & chute heat the air like a 100kW heater?

  9. #9 Rhett Allain
    May 3, 2010

    @Long drop,

    I didn’t even think of that. Great question – I will make it another post when I get a chance.

    thanks for the idea.

  10. #10 Alan Eliasen
    May 25, 2010

    What constants did you assume for area and drag coefficient?

  11. #11 Rhett Allain
    May 25, 2010

    @Alan,

    I assumed a normal sky diver would fall with a terminal velocity of 120 mph and used this to caclulate A and C (with the mass of a normal sky diver).

    Of course, this wouldn’t really be correct since C would probably change around the speed of sound.

The site is undergoing maintenance presently. Commenting has been disabled. Please check back later!