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Calculating the travel time in relativistic travel

  1. Apr 21, 2008 #1
    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
     
  2. jcsd
  3. Apr 21, 2008 #2
    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 ?
     
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