Date: Mon, 23 Nov 1998 19:18:06 -0600
From: "Christopher B. Thrash" 
Subject: Re: Unpowered gliders in VE2 and GT (long)

> On Sun, 22 Nov 1998, Christopher B. Thrash wrote:
> 
> > There should be a condition in which thrust balances
> > drag -- the glider is sinking, but not accelerating -- but the rules
for
> > aDecel don't appear to cover this. 
> 
> 	Yeah, it's called top speed for the applied thrust. 
>  

Okay, I got it to work -- after a fashion.  For standard hulls in GT (very
good streamlining, lifting body option), the answer is

minimum glide angle = 0.00143 Lwt / Sa

where Lwt is loaded weight (in pounds), and Sa is surface area (sf).  Glide
ratio (horizontal distance traveled for vertical distance lost) is then 

glide ratio = (1 - glide angle) / glide angle.

For the ships presented in GT, the actual numbers are:

Ship			dtons	stons	sf	glide ratio	
Iramda			10	58.9	2,000	10.8	
Launch			10	36.0	2,000	18.4	
Lifeboat		10	40.8	2,000	16.1	
Rampart			10	90.3	2,000	6.7	
Gig			20	77.9	3,000	12.4	
Ship's Boat		30	97.1	4,000	13.4	
Fuel Skimmer		40	104.3	5,000	15.7	
Pinnace			40	130.4	5,000	12.4	
Modular Cutter		50	189.6	6,500	11.0	
Interplanetary Shuttle	100	230.0	10,000	14.2	
Shuttle			100	374.5	10,000	8.3	
Suleiman		100	346.8	10,800	9.9	
Suleiman II		100	382.0	10,800	8.9	
Animal			200	398.9	15,800	12.8	
Beowulf			200	631.5	16,600	8.2	
Empress Marava		200	556.6	16,600	9.4	
Akkigish		400	1,340.8	28,200	6.3	
Dragon			400	2,907.4	28,200	2.4

The unweighted average is 11.  Comparable glide ratios from the real world:

High-performance sailplane	25-40
Typical patrol or transport	12-20
High-performance bomber		20-25
Prop-powered trainer		10-15
Jet trainer			9-16
Transonic jet fighter		10-13
Supersonic jet fighter		4-9
Helicopter			3-5

So, the Dragon drops like a rock, and the rest are in the right ballpark
for transatmospheric craft.  For best results, use the aerobraking rules on
VE2, p. 164, to get down from orbit to 15,000 ft and figure the glide from
there.

The derivation follows.

********Gearhead Alert*********
	
All right, let's examine what we have, shall we?  According to VE2:

Top speed (Vmax) depends on aerial motive thrust (Amt) and aerodynamic drag
(Adr), 

Vmax = sqrt (7,500 * (Amt/Adr))				(eqn 1.0, p. 134)

Aerial acceleration (aAccel) depends on aerial motive thrust and loaded
weight (Lwt), 

aAccel = (Amt/Lwt) * 20					(eqn 2.0, p. 135)

Put it another way:

Amt = (aAccel * Lwt) / 20				(eqn 2.1)

But the unpowered glider has no aerial motive thrust and no top speed (Amt
= 0, Vmax = 0), unless it accelerates by diving. This acceleration (a
component of aAccel, in this case the only component) depends on the
fraction of total velocity (V) that is dedicated to vertical movement (Vv,
where V = Vv + Vh; this is only an approximation, but given as such in the
text).  That relationship is

aAccel = (Vv/V) * 20					(eqn 3.0, p. 155)

Combined with equation 2.1, this gives aerial motive thrust and top speed
as

Amt = Vv/V * Lwt					(eqn 2.2)

Vmax = sqrt (7,500 * Vv/V * Lwt / Adr)			(eqn 1.1)

     = (Vv/V)^1/2 * (7,500 * Lwt / Adr)^1/2		(eqn 1.2)

You can see that Vmax varies as a function of Vv/V.  When Vv/V = 1 (a
vertical dive), Vmax equals terminal velocity (p. 157, Falling).  As Vv/V
tends to zero (pure horizontal flight), Vmax also tends to zero until
eventually Vmax is less than stall speed (Vs).  Thus the minimum allowable
value of Vv/V occurs when Vmax = Vs. For the unpowered glider,

Vs = (Lwt / Lift Area) * Sl * Rs			(eqn 4.0, p. 133)

Set Vmax = Vs, and

(Vv/V)^1/2 * (7,500 * Lwt / Adr)^1/2 = (Lwt / Lift Area) * Sl * Rs
							(eqn 5.0)

GURPS Traveller sets some parameters for standard hulls:  Very Good
streamlining, Lifting Body option, no Responsive Hulls.  This allows us to
simplify 5.0 to

(Vv/V)^1/2 * (37,500 * Lwt / Sa)^1/2 = (Lwt / 0.3Sa) * 2.2	(eqn 5.1)

where Sa is hull surface area.  So

Vv/V = 0.00143 Lwt / Sa

for a minimum dive angle to maintain Vmax = Vs.