Acceleration Mass Equation (*LONG*!!)

> Phillip E. Pournelle wrote:

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