Re: Acceleration Mass Equation (*LONG*!!)

Phillip E. Pournelle wrote:
> 
>	  Could someone send me the equation that explains the amount of
mass
> consumed by fuel for continuous acceleration?

I'll try to explain (though I'm far from certain myself...)

Propellant Mass Flow is how many kilograms of fuel/propellant/
reaction mass/whatever that is expended per second.  In rocketry
circles, this is called M-dot (symbolized with a lower case "m" with
a dot over it, alas, this doesn't seem to be on the keyboard)

Mdot	= F / Ve
	= dMp / dT	(duh...)
where
Mdot	= propellant mass flow in kg/sec
F	= Thrust in Newtons or kg m/sec
Ve	= Velocity of exhaust in m/sec
( the following is stupidly obvious, but included for completion )
dMp	= mass of propellant burnt in current burn in kilograms
dT	= duration of current burn in seconds
( in other words, kg/secs = kgs divided by seconds )

How much change in velocity does the rocket experience? This
is called "delta-V", and is measured in meters per second.
( In the Earth Force Source Book, delta-V would determine the
number of movement units the main drive could thrust the ship )

VELOCITY CHANGE OF CURRENT BURN (m/sec)
deltaVb = Ve * 1n[Mbs / Mbe]
where
deltaVb = velocity change of current burn in meters per second
Ve	= Velocity of exhaust in m/sec
1n[x]	= take the natural logarithm of x, (log e) NOT the common
logarithm
Mbs	= Ship's mass at start of current burn in kg
Mbe	= Ship's mass at end of current burn in kg

Now, here are some more equations for your game:

ROCKET EQUATIONS
=======================
UNITS
m	= meter
kg	= kilogram
x^2	= square of x
sqrt[x] = square root of x
g	= 9.81 m/sec^2
1n[x]	= Log(e), natural logarithm of x
G	= 6.67206e-11 Nm^2/kg^2
c	= 299792458 m/sec
p	= 3.141592654
Va	= average velocity (m/sec)
V	= change in velocity (m/sec)
Vi	= initial velocity (m/sec)
Vf	= final velocity (m/sec)
S	= change in distance (m)
T	= time (seconds)
A	= acceleration (m/sec^2)
Ai	= "instantaneous" acceleration (m/sec^2)
ENGINE PARAMETERS
F	= Thrust (Newtons or kg m/sec)
Pw	= Thrust Power (kW)
Isp	= Specific Impulse (seconds)
Mdot	= Propellant mass flow (kg/sec)
Ve	= Velocity of exhaust (m/sec)
Mps	= Mass of propulsion system (power plant+thrust system) (kg)
dMp	= Mass of propellant burnt in current burn (kg)
Mp	= Total mass of propellant carried (kg)
Alpha	= Specific Power = Pw / Mps (kW/kg)
Vch	= Characteristic Velocity
Epsilon = percentage of propellant mass converted into energy
VEHICLE PARAMETERS
Mpl	= Mass of ship's payload (kg)
Ms	= Ship's structural mass (kg)
Mt	= Ship total mass = Mp + Mpl + Mps + Ms (kg)
Me	= Ship's mass empty (i.e., all propellant burnt)  (kg)
	= Mt - Mp
Mc	= Ship's "current" mass (at this moment in time) (kg)
Mbs	= Ship's mass at start of current burn (kg)
	{At start of mission = Mt. Later it is Mt - (sum of all DMp's of
all
burns)}
Mbe	= Ship's mass at end of current burn (kg)
Lambda	= Ship's mass ratio = Mt / Me
deltaV	= Ship's total velocity change capability (m/sec)
dTm	= Maximum duration of burn (seconds)
MISSION PARAMETERS
deltaVb = Velocity change of current burn (m/sec)
dT	= Duration of current burn (seconds)

* WARNING * The below equations assume a constant acceleration,
which is not  true for a ship expending mass (for instance, 
propellant). Ai = F/Mc so as the ship's mass goes down, the acceleration
goes up.
============================================
When you have two out of three of average velocity (Va),
change in distance (S) or time (T)
Va = S / T
S = Va * T
T = S / Va
============================================
When you have two out of three of acceleration (A),
change in velocity (V) or time (T)
A = V / T
V = A * T
T = V / A
============================================
When you have two out of three of change in distance (S),
acceleration (A), or time (T)
plus Initial Velocity (Vi)   Note: if deaccelerating, acceleration A is
negative
S = (Vi * T) + ((A * (T^2)) / 2)
A = (S - (Vi * T)) / ((T^2) / 2)
T =  (sqrt[(Vi^2) + (2 * A * S)] - Vi) / A
If Vi = 0 then
S = (A * (T^2)) / 2
A = (2 * S) / (T^2)
T = sqrt[(2 * S) / A]
============================================
When you have two out of three of change in distance (S),
acceleration (A), or final velocity (Vf)
plus Initial Velocity (Vi)   Note: if Vf < Vi, then A will be negative
(deacceleration)
S = (Vf^2 - Vi^2) / (2 * A)
A = (Vf^2 - Vi^2) / (2 * S)
Vf = sqrt[Vi^2 + (2 * A * S)]
If Vi = 0 then
S = (Vf^2) / (2 * A)
A = (Vf^2) / (2 * A)
Vf = sqrt[2 * A * S]
============================================
If the ship constantly accelerates to the midpoint, then
deaccelerates to arrive with zero velocity at the 
destination:
T = 2 * sqrt[S / A]
S = (A * (T^2)) / 4
A = (4 * S) / (T^2)
 ============================================
THRUST (Newtons or kg mt/sec)
F	= Mbs * A
	= Mdot * Ve
	= Mdot * g * Isp
	= (dMp * Ve) / dT
============================================
THRUST POWER (kW)
Pw	= (Mdot * (Ve^2)) / 2
Pw	= (dMp * (Ve^2)) / (2 * dT)
============================================
SPECIFIC IMPULSE (seconds)
Isp	= Ve / g
	= F / (g * Mdot)
============================================
PROPELLANT MASS FLOW (kg/sec)
Mdot	= dMp / dT
		= F / (g * Isp)
		= F / Ve
============================================
VELOCITY OF EXHAUST (m/sec)
Ve	= g * Isp
		= F / Mdot
Ve/c	= sqrt[ epsilon * (2-epsilon)]
Ve/c	= exhaust velocity in fractions of the velocity of light
============================================
MASS OF PROPELLANT BURNT IN CURRENT BURN (kg)
dMp	= Mdot * dT
		= (F * dT) / (g * Isp)
		= (F * dT) / Ve
============================================
SPECIFIC POWER (kW/kg)
alpha	= Pw / Mps
============================================
CHARACTERISTIC VELOCITY
Vch	= sqrt[ 2 * alpha * dT ]
============================================
SHIP'S TOTAL MASS (kg)
Mt	= Mp + Mpl + Mps + Ms
============================================
SHIP'S MASS EMPTY (all propellant burnt) (kg)
Me	= Mt - Mp
============================================
SHIP'S MASS AT END OF BURN (kg)
Mbe	= Mbs - Mbp
============================================
 SHIP'S MASS RATIO (dimensionless number)
Lambda = Mt / Me
============================================
SHIP'S TOTAL VELOCITY CHANGE CAPABILITY (m/sec)
deltaV	= Ve * 1n[Lambda]
	= g * Isp * 1n[Lambda]
============================================
MAXIMUM DURATION OF BURN (seconds)
dTm	= Mp / Mdot
============================================
VELOCITY CHANGE OF CURRENT BURN (m/sec)
deltaVb = Ve * 1n[Mbs / Mbe]
============================================
ACCELERATION (m/sec^2)
A	= F / Mc
	= (Mdot * Ve) / Mc
	= (Mdot * g * Isp) / Mc