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?
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:
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):
Maybe that is not the best model, but I am sticking with it.
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)
I hinted above, but just to be clear, I am using the more realistic expression for gravity:
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).
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):
And if I use the speed of sound at sea level:
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.
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.
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.