Monday, October 09, 2006

*THINKING ABOUT IT*

****THINKING ABOUT IT****
Equations


Einstein loved thought experiments.

1. Accelerate for a Day
Perhaps one of Einstein's more famous thought experiments describes an elevator which accelerates at same rate as a free falling object near Earth's surface. (This acceleration rate is known as "g", approximately 10 meters per second per second (10 m/sec2)). Einstein concluded the occupants could not discern whether that elevator was static on Earth or g-force accelerating in space.

Keying on g, express same g value in different time/distance dimensions.
SUMMARY
g 10 m/sec

sec
864 km/sec

day
0.5 AU/day

day
0.289%c

day
Equivalent Expressions of g

2. Flight Profile To the Planets
We could leverage Einstein's thought experiment to routinely travel at g-force to neighboring planets in days/weeks.

Humans have sent a few probles on interplanetary trips, but these trips have always been at orbital velocity. There have been a few ingenious tricks of increasing velocity such as "gravity assists" from mid-trip planets as well as ingenious economies from Hohman Transfers. Even so, interplanetary trips now take months if not years; they still must be accomplished in zero-g and other uncomfortable conditions. In short, interplanetary trips are now impractical for anything other than Artificial Intelligence (AI) devices.


On the other hand, g-force travel could make interplanetary travel comfortable for humans. Constant acceleration would impart very high speeds to the spacecraft; so high that the entire second half of the flight would be needed to decelerate the vessel back to orbiting speed at destination planet.

Typical g-force Flight Profile Example Let d = trip distance = 10 AU; express g as 0.5 AU/day2.
dAcc=d/2

tAcc=(2dAcc/g)= (d/g)
tDec=(d/g)=tAcc
tAcc+tDec=2√(d/g)=tTtl
Acceleration
Distance
Acceleration
Time
Deceleration
 Time
Total Trip
Time
dAcc=10 AU

2

=5 AU
tAcc= √(10 AU

.5AU/dy2)
)=4.47dy
tDec=tAcc=4.47dy
tTtl=2×4.47dy=8.94dy

3. Momentum Makes It Happen
Momentum works for current technology rockets when high velocity gas exits quickly in one direction and rocket increases speed in opposite direction.

 

MOMENTUM EXCHANGE
mfuel* vExh = Mship * Vship
mfuel : Mass of fuel consumed.
vExh: Velocity of fuel particles exiting exhaust.
Mship : Mass of Spaceship.
Vship : Velocity of Spaceship.
Rocket Exhaust = Ship Propulsion
CONVERT TO RATES














On each side of equation,


 
 
...divide selected term by one second.
DEFINE NEW TERMS













ffsec =





5. Finite Range
Arbitrarily assume ship's initial Gross Weight (GW) of 100,000 metric Tonnes (mT).

With a daily consumption rate of Δ = 1%; then, that day's fuel consumed would be 1,000 mTs. Thus, we could readily determine fuel requirements if we only knew ship's mass.

However, we don't really need to compute masses of ship or fuel loads. Since Daily Difference, Δ, is a percentage of GW,
mfuel













1 sec
Fuel flow per second:
 
Select: Vship = 10 m/sec
ffSec* vExh = MShip * g

Thought experiment assumes same principle works for a spaceship and constant stream of very high speed, plasma particles (ions).
4. Mass to Motion
Planes, trains, and automobiles (not to mention people) routinely convert mass (aka "fuel") to energy and thus to motion.
Δ=ffday

MShip
=86,400 ffSec/day

30.65 * √(mr2-1)* 106xffsec
=0.282%/day

(mr2-1)
Thought experiment makes some initial assumptions to determine a typical daily difference in vehicle's gross weight due to fuel consumption.
we can readily approximate fuel requirements for a given flight duration. (Thought experiment uses aeronautical term, Percent Take-Off GW, %TOGW.)
Consumption Rate.
Approximate fuel requirement by multiplying duration times consumption rate.













%TOGW=Δ * t
Exponentials.
Exponentials can quickly determine more accurate fuel requirements for specified flight durations.
%TOGW=1- (1-Δ)t
Logarithms.
Logarithms can determine range for specified fuel load.

tR = log(1-%TOGW)

log(1-Δ)
=log(%GWFin)

log(1-Δ)
Thought experiment artificially assumes 100% efficiency for interplanetary flights because momentum of high speed (% light speed) exhaust particles provides so much propulsion capability that realistic efficiencies can be disregarded. (Not so for interstellar flights which will be discussed in Volume II.)
 

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