# Re: New idea Laser launcher/scoop systems

```>>Kelly, how did you calculate these numbers?
>>
>>Tim
>
>found the equation.
>
>A.1 The rocket equation
>
>The "rocket equation" tells you that
>the ratio of rocket mass when full of fuel to rocket mass after
>burning the fuel is:
>
>          m(i) / m(f) = exp ( dV / v(e) )

Yups, I already found it when you pointed out where to look for it.

===============================================================================

>>I already see what makes the difference (besides the fact that formula
>>doesn't take into account what the energy:mass ratio of the fuel is). For
>>slightly different exhaust speeds (around 0.08c) the ratios are optimal. I
>>must have overlooked that in Kellies table, where at the top this speed of
>>25M is used. I don't understand however why all the other numbers are shown
>>in the table.
>
>I wanted to show all mass ratio's with various specific impulse numbers.

OK, unfortunately (as you stated below) the paper didn't show all there was,
your numbers only showed a decrease of mass for lower exhaust-velocities.
I found out that for high exhaust speed the mass goes up because of the
extra kinetic energy (could be overcome by beaming) and for lower speeds the
mass goes up too, because of the waste of energy (ie. using fusion fuel as
reaction mass).

So one needs to find the valley between the (ever rising) hills.
The formula to do this:

BestG:=1+1/(f-1);                     (the relativistic gamma)
BestV:=c*Sqrt(Sqr(BestG)-1)/BestG     (the accompanying velcity)

The optimum exhaust-velocity is only dependent on the mass:energy ratio of
the fuel (thus not on the final velocity).

f   exhaust velocity (in c)
200        0.09987
250        0.08935
300        0.08158
350        0.07554
400        0.07067
450        0.06663
500        0.06321

(f is the mass:energy ratio (about 270 is the best for fusion untill now))

The next table shows the ship:fuel ratios needed to accelerate upto the
final velocities that vary horizontally. Vertically the energy:mass ratios
of the fuel are varied.
Every time the optimal exhaust speed is used (this can be looked up in the
previous table).

End velocity -->
0.10      0.20      0.30      0.40    0.50    0.60    0.70
200     2.7       7.6      22.2      69.5     244    1032    5906
250     3.1       9.7      31.9     114.6     467    2338   16422
300     3.4      12.0      44.4     180.0     839    4896   41401
350     3.8      14.6      60.2     272.7    1439    9662   96907
400     4.1      17.6      79.8     401.4    2376   18191  213876
450     4.5      21.0     104.1     577.2    3805   32958  449882
500     4.9      24.7     133.8     813.9    5941   57820  908988
^
+--- Energy:mass ratio of the (fusion) fuel

If anything is unclear, don't hesitate to ask.

Timothy

```