Free fall of person in a sled to direct horizontal motion by atmospheric resistance

[Dukon]

Let there be a person in a not yet optimally designed sled at h meters in height. Let this sled free fall but user can steer by tilting their body weight in the sled or by optimal sled shape design point it in some horizontal direction where it is wanted to go - in any horizontal direction but once picked fixed. How to calculate horizontal distance d achievable as function of height h. Thus what is f(h) = d.

Put another way, imagine a helicopter rises to a height h, but then shuts off all engines and power, and just free falls. Imagine also the bottom of chopper is a strategically designed shape engineered to maximize horizontal force imparted to the vehicle such that it lands a distance d away from the direct vertical drop from where it started its free fall.

What is involved in calculating just this free fall + atmosphere resistance, assuming no wind as initial approximation. If theory is suggested plz provide calculation with estimated values for required parameters, not just a howto but also a numerical answer with assumed values.

Assume sled is a square. Then a rectangle. Then a bowl of spherical shape. Next of some prolate spheroidal shape. Also, a perhaps arbitrary shape determined by optimizing and maximizing d given h. Note that atmospheric properties are functions of h.

 

[Baluncore]

What is involved in calculating just this free fall + atmosphere resistance, assuming no wind as initial approximation. If theory is suggested plz provide calculation with estimated values for required parameters, not just a howto but also a numerical answer with assumed values.

The sled will have an optimum glide ratio, and that will determine the range from the initial place of fall.

What is the maximum height at the beginning of the fall? Is it outer space?
Do you assume the atmosphere is standard at sea level, or allow for variation of the pressure and density with height?

 

[Dukon]

Thank you for commenting
h can be set to anything we want, lets make it 10m, 100m, 1000m, 1500m, so we get a start of an initial table of h, d values. I just want to know what d is given h, and then find the h that drops us from NYC to Chicago, examples like this...NYC to LA, NYC to London UK etc -- assume in future it is easy/free to go vertically up. Can we just drop from NYC to LA (in which case travel time must be rather fast compared to current technology)

Note atmospheric properties are functions of h, so we can ignore such effects at first just to get the first approximation without them completed first. Then once this is known, if this is clear by itself and gives us initial answers for the h,d table we can add in variations of atmospheric properties with h as a later improvement

But what is optimal sled shape? square, rectangle, bowl of what shape, etc. Does shifting mass to one side actually deliver horizontal force as envisioned, etc

 

[Baluncore]

But what is optimal sled shape?

A sailplane or glider.

The height required for an intercity glide is higher than the atmosphere, so the initial free fall, without air, will limit the maximum range.

 

[Baluncore]

The best possible glide ratio of any device will be about 70:1 for a sailplane. The available depth of the atmosphere is less than 50,000 feet. The maximum range must therefore be less than 70 * 50,000 ft = 3,500,000 feet = 1067 km, which is less than the distance from NYC to Chicago.

 

[Dukon]

Thank you highly for being specific with numerical results. Very helpful. Much appreciated!

 

[nasu]

Isn't the glide ratio dependent on the horizontal velocity? The OP seems to assume that there is no initial horizontal velocity, doesn't it?

 

[A.T.]

Isn't the glide ratio dependent on the horizontal velocity? The OP seems to assume that there is no initial horizontal velocity, doesn't it?

If you start out falling vertically, you will obviously lose some altitude before you can start gliding optimally. But for the rough estimate of the upper range bound one can neglect that.

 

[nasu]

If you start out falling vertically,

In a sled??

 

[russ_watters]

In a sled??

No, a "sled".

Also, in case your question wasn't addressed; optimal glide ratio is a function of aerodynamic efficiency only. A heavier "sled" will glide at a higher speed, but the same angle.

 

[A.T.]

In a sled??

... a strategically designed shape ....

 

[DaveE]

Isn't the glide ratio dependent on the horizontal velocity?

These specifications like glide ratio, climb rate, etc. all assume you are operating the aircraft to maximize that particular performance metric or, like fuel burn rate, they have specified operating parameters. You can always make it worse, but usually not better.

 

[Baluncore]

The OP seems to assume that there is no initial horizontal velocity, doesn't it?

Yes, but what is required to quickly convert vertical height into horizontal velocity, is the second half of an Immelmann turn.
https://en.wikipedia.org/wiki/Immelmann_turn
That manoeuvre must be done in air that is sufficiently dense for the control surfaces to operate without stalling into a spin. The altitude must be less than "coffin corner" for a high-altitude glider, such as the U2. There is a diagram here.
https://en.wikipedia.org/wiki/Coffin_corner_(aerodynamics)
The U2 can creep up into coffin corner from below, by using a careful search, but falling into coffin corner from above is close to impossible. The diagram provided for the U2, shows it might be done below 76,000 feet, and at about 80 knots. My guess at an upper limit of 50,000 feet was based on the operating altitude of business jets, they are engineered to be stable, to not stall or spin at that altitude. Maybe some U2 pilot has, after a flameout, survived an Immelman turn between 50 and 76 thousand feet.

The OP's aerodynamic "sled" concept seems to morph into a competition glider, maybe launched vertically with its wings swept back, that then transforms into an optimum sailplane at the crest of its launch.

Watch the critical drop at the start of some of the "Birdman Rally" competitions, where teams build an aircraft, then step, walk, or run off a platform, with the aim of gliding to the furthest possible range.