MASS TO MOTION
Converting mass to motion
is a simple fact of life.
is a simple fact of life.
We do it with planes,
trains and automobiles.
trains and automobiles.
Robert Hutchings Goddard (October 5, 1882August 10, 1945). Throughout his extremely productive life, Goddard realized the potential of missiles for space flight and greatly contributed to practical realization. In 1914, Goddard received his first two U.S. patents.
Altogether, Robert Goddard developed 214 patents; many were awarded posthumously.
In Goddard's" autobiographical essay: "on the afternoon of October 19, 1899, I climbed a tall cherry tree and ….. looked towards the fields at the east, I imagined how wonderful it would be to make some device which had even the possibility of ascending to Mars ...."
He died on August 10, 1945, four days after the first atomic bomb was dropped on Japan. Expanded Text 
Most people routinely buy fuel for their automobiles. We all know the purpose of gasoline is to enable vehicles to travel; thus, the fuel (mass) converts to energy which powers the engine which moves the car down the road (motion). Stop refueling, and the gas gauge eventually drops to zero; thus, we intuitively know that fuel consumption is another fact of life.
%TOGW = Fuel/M_{Ship}
Of course, aircraft flight also depends on fuel; some aircraft designers use the term, Percent Take Off Gross Weight (%TOGW), to indicate portion of aircraft's initial gross weight needed for fuel at beginning of flight.
EXAMPLE: At beginning of trip, let a ship's total weight be 100 tons with 60 tons of fuel, then
%TOGW = 60 tons/100 tons = 60%
for that particular trip. With a %TOGW of 60%, ship must limit all other mass (infrastructure, payload) to 40% of ship's mass. Since payload is the purpose of any trip, we want to minimize %TOGW.
Basic Assumptions:
In free space (no air resistance), an object will travel at a constant velocity until a force acts upon it. Let our thought experiment's spaceship eject particles for an entire second; the particle momentum (collective mass times their velocity) will force the spaceship to react accordingly by changing it's velocity.
Thought experiment assumes that collective exhaust particle momentum throughout a second can be described by one vector (direction and length); thus, thought experiment further assumes an equal and opposite vector describes spaceship's momentum for same duration (one second).
Thought experiment's main premise: spaceship travels at gforce throughout entire voyage; thus, the vessel's velocity must increase by 10 m/sec for every second. Of course, this value approximates acceleration rate of free falling objects near Earth's surface. According to Einstein's famous thought experiment, occupants will feel same weight sensation as if they were static on Earth's surface. Thus, we're constrained to use the value 10 m/sec for the term, V_{Ship}. Since we specify that value for every second of flight, we can substitute g (=10 m/sec^{2}) for V_{Ship}/sec.
 
Progressive Elaboration
Following series of tables will be modified in constructive ways.
Independent Variable (IV). Thought experiment (TE) presumes that humanity will learn to design and construct particle accelerators onboard interplanetary space vessels for the purpose of expelling continuous stream of high speed exhaust particles. This particle stream will be of sufficient quantity to propel the space vessel at continuous gforce. TE further presumes continual improvement of these onboard particle accelerators such that the particle exhaust velocity (V_{Exh}) will continually increase as time goes on. Thus, first table will independently vary V_{Exh} in a range of values. Subsequent tables will show V_{Exh} in higher ranges.
Dependent Variable (DV). Previous work shows that M_{Ship} depends on V_{Exh}. Thus, TE further presumes that as V_{Exh} increases, that initial results will see a corresponding increase in ship size, M_{Ship}, capability. Consequently, this will decrease the %TOGW value.
Fuel. For each second of space flight, a collective mass of fuel is consumed; then, transformed into ions and accelerated into near light speed exhaust particles. Since this is a Thought Experiment, we can choose a quantity which best helps us draw useful conclusions. Thus, we initially choose a value of one kilogram of mass (m_{fuel}) for every second of powered flight. Furthermore,let us express this mass per second (m_{fuel}/sec) as fuel flow per second (ff_{sec}).
ff_{Day} = 86,400× ff_{sec }= 86,400 kg/day == 86.4 mT/day
With trip time, t, in days, determine total fuel, F, for trip:
F = ff_{Day} × t
Finally, this lets us compute percent Take Off Gross Weight, %TOGW, portion of ship's initial gross weight needed for fuel.
%TOGW = F/M_{Ship}

 

 
Recall Effects of Constant Acceleration
Let d equal straight line distance from departure to destination.
d = 1/2 × g × t^{2} Solve for t: t = √(2 d / g ) Let g be same acceleration as experienced in free fall near Earth's surface. For convenience, assume g closely approximates 10 m/sec^{2} Straightforward conversions allow us to further assume g approximates 0.5 AU/day^{2}. EXAMPLE: Let d=0.4 AU, then calculate trip time for constant acceleration for entire distance: t = √(2 × 0.4 AU / 0.5 AU/day^{2}) = 1.26 days
One can compute travel distance with trip time as independent variable. Thus, solve for d:
d =(1/2) × g × t^{2}
EXAMPLE:
Accelerate at 0.5 AU/day^{2} for .9 days; then, determine distance traveled: d = 0.5 × 0.5 AU/day^{2} × (.9 day)^{2} = 0.202 AU 
WARNING!!!!!!!!!!!!!
After only one day of gforce acceleration, spacecraft attains enormous velocity: v_{Fin} = 864 kilometers per second! Assume velocity increases 10 m/sec for every second of gforce propulsion. v_{Fin }= t_{sec} × g = 86,400 sec ×10 m/sec² v_{Fin }= 864,000 m/sec = 864 km/sec STATE ANOTHER WAY. For every day of constant acceleration, g; spacecraft velocity increases 864 km/sec: EXAMPLE: After two days of gforce spaceflight: v_{Fin} = g × t = 864 km/sec /day × 2 days = 1,728 km/sec To accommodate these enormous speeds interplanetary gforce space travel needs FLIGHT PROFILE: Slowdown at midpoint. 
 
Mix Units.
To conveniently avoid large cumbersome numbers.
Mix velocity units:
Thus, express total fuel, F, in mT, metric Tonnes.

Adjust Flight Profile
A better flight profile would accelerate to halfway then slowdown. Thus, expression for total time of travel changes to
t = 2√(d/g)
which is explained further below. To accommodate enormous velocity gains and still maintain gforce throughout the entire flight, accelerate at gforce until the halfway point; then, decelerate at gforce for remaining half of time/distance. Consider it axiomatic that the two halves of the flight duration are equal. The main difference between these two flight phases is that the ship's propulsion vector will point in opposite directions.

Exhaust particles gain enormous velocity, V_{Exh}, prior to exiting spacecraft.
V_{Exh} is a significant portion of c, speed of light.
Thus, V_{Exh} adds enormous momentum to propulsion particles as they exit the spacecraft.
Thought experiment further assumes that relativistic mass growth imparts even more momentum which adds even more mass capacity to the spaceship. This growth can be quantified by the Lorentz Transform.
LORENTZ TRANSFORM
Compare exhaust particle's original mass (m_{O}) with increased relativistic mass (m_{r}).


Let m_{O }= 1; then,


Substitute d_{c} × c for v_{Exh};
then, ② reduces to: 

Let m_{O }= ff_{sec}
Let original mass be the fuel consumed at rest. To indicate this quantity, TE uses "ff_{sec}"_{, }fuel flow per second. To indicate mass of exhaust particles per second, TE uses "ff_{Exh}".
Show relativistic growth of fuel particles; consumed fuel, ff_{sec}, becomes high momentum
exhaust particles, ff_{Exh}. 

Expressed another way: ff_{Exh} = m_{r} × ff_{sec}
Example: Let V_{Exh}= d_{c }× c = .3 c
m_{r }=1.0/√(1  d_{c }^{2}) = 1.0/√(1.3^{2})
ff_{Exh }= m_{r} × ff_{sec} = 1.048 ff_{Sec}
Example: Let V_{Exh}= d_{c }× c = .3 c
m_{r }=1.0/√(1  d_{c }^{2}) = 1.0/√(1.3^{2})
ff_{Exh }= m_{r} × ff_{sec} = 1.048 ff_{Sec}
To account for relativistic effects on exhaust particles, momentum exchange equation should use ff_{Exh}.


Make suitable substitutions:
v_{Exh} = d_{c} × c ff_{Exh} = m_{r} × ff_{sec} 

c ≈ 300,000,000 meters/sec
g ≈ 10 meters/sec/sec 
Express ff_{sec} in kg; M_{ship }in mTs;
Conversion: 1kg = 10^{3} mT 
Replace c/g with 30,000
M_{ship}  = 
m_{r}×ff_{sec}×d_{c}×30,000


d_{c}  ff_{Exh}  M_{ship} 

.1

1.005 kg

3,015 mT

.2

1.021 kg

6,124 mT

.3

1.048 kg

9,435 mT

v_{Exh}/c

1 kg/√(1d_{c}²)

ff_{Exh} × d_{c} × 30,000

distance  Given  (d)  5 AU  10 AU  15 AU  20 AU 
20 AU range includes orbits of Jupiter and Saturn
as well as large portion of orbit of Uranus. 

time  4√(d/g)  (t)  12.65days  17.89days  21.91days  25.30days 
To accommodate two way trip,
double the time factor from 2√(d/g) to 4√(d/g). 
Fuel  ff_{Day}×t  (F)  1,093 mT  1,546 mT  1,893 mT  2,186 mT 
ff_{Day }= 86.4 mT /day

v_{Exh}(d_{c}×c)  ff_{Exh}  M_{Ship}  %Take Off Gross Weight (%TOGW) 
2Way Flight Profile:
assumes four phases of equal duration
as described below.
Phase 1: Accel from dept to midway, t= √(d/g).
Phase 2: Decel from midway to dest, t= √(d/g).
Phase 3: Accel from dest to midway, t= √(d/g).
Phase 4: Decel from midway to dept, t= √(d/g).
 
.3c  1.048kg 
9,435 mT
 11.59%  16.39%  20.06%  23.17%  
.4c  1.091kg 
13,093 mT
 8.35%  11.81%  14.46%  16.70%  
.5c  1.155kg 
17,321 mT
 6.31%  8.93%  10.93%  12.62%  
.6c  1.250kg 
22,928 mT
 4.77%  6.74%  8.26%  9.53%  
.7c  1.400kg 
29,965 mT
 3.65%  5.16%  6.32%  7.30%  
.8c  1.667kg 
40,760 mT
 2.68%  3.79%  4.64%  5.36%  
Given  ff_{Sec} √(1d_{c}^{2})  ff_{Exh}×d_{c}×30,000  %TOGW = F / M_{Ship} 


%TOGW depends on m_{r}, relativistic growth factor and time, t. As shown in above equation, %TOGW is inversely proportional to m_{r}. As it grows, %TOGW shrinks which means less matter needs to convert to energy for same days performance. Thus, more of ship's GW can be devoted to payload, a good thing. Father of Modern Rocket Propulsion Sidebar: Robert Goddard 
Electric Propulsion
RHG's Early Career. As a young academic, he divided his time between official research on electricity and his personal passion for propulsion. This led him to Electric Propulsion (EP). In modern times, we wonder why was Goddard more concerned with the electrostatic acceleration of electrons rather than ions? (Recall that ions are much more massive than electrons; thus, they contribute far more momentum between the rocket and exhaust particles.) An online paper suggests five reasons:

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