# Re: Comparison of Sail vs. RAM (Was Laser Aperture Size)

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Tim:
> Kevin:
> >have no clue as to how to find it (aside from trial and error which is
> >what I intend to do this weekend)
>
> Since you probably don't have the right formulas trial and error will not
> work either, finding these may indeed by a bit nasty. Since I've done such
> calculations before, it will be a bit easier for me.
>
> >Any help will be greatly appreciated..
>
> Well, I suppose you want to find the solution that consumes the least energy
> (not necessary the least reaction mass).
>
> So we need to optimize the energy.
>
> Energy needed to decelerate a ship depends on it's empty mass, its start
> velocity, the exhaust velocity.
>
> E(Mo,Vstart,Vexh)
>
>                   1
>          ------------------
>                         2
> (1)  g =            Vexh
>           Sqrt[1 - ------]
>                      2
>                     c
>
>              M[t] a
> (2)  j[t] = --------
>              g Vexh
>
>           c           Vstart
> (4)  T = --- ArcTanh[--------]
>           a             c
>
>                  T
>                  /
> (5)  M[t] = Mo + | j[t] dt
>                  /
>                  t
>
>                           T
>                     a     /
> (6)  M[t] = Mo + -------- | M[t] dt
>                   g Vexh  /
>                           t
>
> Solving this differential equation gives:
>
>                     a (T - t)
> (7)  M[t] = Mo Exp[-----------]
>                      g Vexh
>
>                   T
>            2      /
> (9)  Ek = c (g-1) | j[t] dt
>                   /
>                   0
>
>            2                a T
> (10) Ek = c (g-1) Mo Exp [--------]
>                            g Vexh
>
>            2                 c              Vstart
> (11) Ek = c (g-1) Mo Exp [-------- ArcTanh[--------]]
>                            g Vexh              c
>
> Finally... E(Mo,Vstart,Vexh)
>
>
> Now we only need to find its minimum, that's easy: solve dE/dVexh=0
>
> Well, that's not easy but the minima are:
>
> Vstart  Vexh optimal  Fuel:ship-ratio  Energy per kg of ship (in Joules)
>  0.1        0.05           7.4            8E14
>  0.2        0.10           7.4            3E15
>  0.3        0.15           7.3            8E15
>  0.4        0.21           7.2            1E16
>  0.5        0.27           7.1            2E16
>  0.6        0.34           6.9            4E16
>  0.7        0.41           6.8            6E16
>  0.8        0.51           6.4            9E16
>  0.9        0.64           5.9            2E17
>  0.99       0.87           4.5            4E17
>  0.9996     0.957          3.6            8E17
>
But does this take into account the push from the maser?

I think I understand what you did, but I am not at all sure that I
comprehend the details.

assuming that we went on a continuous 1 G thrust profile, we should be
near .9905 at turn around (your Vstart?) then optimal Vexh is .87 and
RM to ship ratio is 4.5 to 1,  and the masers have to provide
4E17 Joules /sec/Kg = Watts/Kg ????!

I really really want to thank you for taking the time to do the
calculation.  but could you please explain the result?

Kevin

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