# Calculating the travel time in relativistic travel

A spaceship with empty mass MSS(?kg), start its journey from A to B which distance is RAB(?ly). The amount of fuel the spaceship
initially carry is MFUEL(?kg), which energy per mass is EPM (?Joule/kg). The spaceship engine efficiency is SEP (?%), and the engine is capable of burning FPS (?kg/s) mass of fuel per second. If the interstellar medium friction constant is IMD (?/s), calculate :

1. The maximum cruising speed ? (SRF and IRF)
2. Length of acceleration and deceleration phase ? (SRF and IRF)
3. Length of constant cruising speed ? ( I expect it to be harder if
there is IMD ). (SRF and IRF)
4. The heat generated by friction ?
5. How the question may look like if we add the energy required to
support the crew which is ESC (Watt) into the problem ?

SRF = Ship's Reference Frame
IRF = Inertial Reference Frame

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Here is where my math work had progressed so far :

Look in the variable definition above, in order to understand what
each variables in the equations below means.

So SSP = SEP*EPM*FPS ... (1)

Since (SSP=Space Ship Power) is defined as, the amount of energy the
space ship is capable of produce per unit of time, and energy means
the capability to move a mass M as far as H, using certain amount of
acceleration A, that means :

SSP = TMAR*A*H/dt = 0.5*TMAR*A^2*t ... (2)

TMAR : Total Mass After Relativity.

Which is defined as :

TMAR = TMBR/sqrt(1-(v/c)^2) ... (3)

TMBR = MSS + MFUEL - FPS*t ... (4)

TMAR : Total Mass Before Relativity.

So, I subtitute SSP from (1) to (2) :

SEP*EPM*FPS = 0.5*TMAR*A^2*t , move A^2 from right hand side to left
hand side :
A^-2*SEP*EPM*FPS = 0.5*TMAR*t, move SEP*EPM*FPS from left hand side
to right hand side :

A^-2 = (0.5*TMAR*t) / (SEP*EPM*FPS) , flip both side

A^2 = (SEP*EPM*FPS) / (0.5*TMAR*t) ... (5)

Then I subtitute TMBR from (4) to (3) :

TMAR = (MSS + MFUEL - FPS*t)/sqrt(1-(v/c)^2) ... (6)

Then I subtitute TMAR from (6) to (5) :

A^2 = (SEP*EPM*FPS) / (0.5*(MSS + MFUEL - FPS*t)/sqrt(1-(v/c)^2)*t)

A^2 = (SEP*EPM*FPS* sqrt(1-(v/c)^2)) / (0.5*(MSS + MFUEL - FPS*t)*t)

A = sqrt((SEP*EPM*FPS* sqrt(1-(v/c)^2)) / (0.5*(MSS + MFUEL -
FPS*t)*t))

Since A = dv(t)/dt, the problem become this differential equation :

dv(t)/dt = sqrt((SEP*EPM*FPS* sqrt(1-(v(t)/c)^2)) / (0.5*(MSS + MFUEL
- FPS*t)*t))

Written down in Maple Format :

diff(v(t),t) = sqrt((SEP*EPM*FPS* sqrt(1-(v(t)/c)^2)) / (0.5*(MSS +
MFUEL - FPS*t)*t));

So, did I do something wrong ?
Where should I put the IMD (Interstellar Medium Friction Constant (?/s)) into the equation ?
Is there a way to integrate the differential equation ?