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# the Destroyer future - a new take, with fission!

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Cheatsheet for the relativistic rocket (was Re: the Destroye by Bill Woods   » Sun Jul 12, 2015 8:55 am

Bill Woods
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Posts: 571
Joined: Tue Jun 11, 2013 11:39 am

Jonathan_S wrote: Since RFC seems to ignore relativity for wedge acceleration, we can just use Newtonian formulas for all this.
I didn't, since 0.3c is mildly relativistic. Here's the set of formulas I've got:

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t, d, v, a are measured by the moving observer.
T, D, v, A are measured by a 'stationary' observer.
From a standing start, so at t_0 = T_0 = 0 s, d_0 = D_0 = 0 m and v_0 = 0 m/s;
a is a constant.

Newtonian rocket*: v(t) /c = at/c, while,
Einsteinian rocket: v(t) /c = tanh(at/c)
http://math.ucr.edu/home/baez/physics/R ... ocket.html

* Though a 'rocket' is an unlikely vehicle to be producing constant acceleration for long periods of time.

With x == at/c
gamma(x) = 1/sqrt{1 - [v(t)/c]^2} = cosh(x)

t(x) = c/a x
d(x) = c^2/a [1 - 1/gamma(x)] = c^2/a [1 - 1/cosh(x)]
v(x) /c = tanh(x)

T(x) = c/a sinh(x)
D(x) = c^2/a [gamma(x) - 1] = c^2/a [cosh(x) - 1]
A(x) = a /gamma(x)^3 = a/cosh^3(x)

With y == aT/c = sinh(x)
gamma(y) = sqrt{ y^2 + 1 }

t(y) = c/a arsinh(y)
d(y) = c^2/a [1 - 1/gamma(y)] = c^2/a [ 1 - 1/sqrt{ y^2 + 1 } ]

T(y) = c/a y
D(y) = c^2/a [gamma(y) - 1] = c^2/a [ sqrt{ y^2 + 1 } - 1 ]
v(y) /c = y /gamma(y) = 1/sqrt{ 1 + 1/y^2 }
A(y) = a /gamma(y)^3 = a { y^2 + 1 }^-3/2

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Setting a = 5000 m/s2, and plugging in v/c = 0.3, I got
T = 18,900 s = 5.24 hr,
D = 868 e9 m = 48.3 lt-min.

Setting a = 6000 m/s2,
T = 15,700 s = 4.36 hr,
D = 723 e9 m = 40.2 lt-min.

Jonathan_S wrote: (Now in actuality I didn't work it out like this until now, I simply plugged the numbers into a spreadsheet I already had for missile accelerations; treating the ship like a very slow, very long endurance, missile)
Just what I did! If you compare, you'll see the author's error isn't plot-critical; missiles run slower, but they run longer so they actually have longer ranges than advertised.
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Imagined conversation:
Admiral [noting yet another Manty tech surprise]:
XO, what's the budget for the ONI?
Vice Admiral: I don't recall exactly, sir. Several billion quatloos.
Admiral: ... What do you suppose they did with all that money?
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Re: Cheatsheet for the relativistic rocket (was Re: the Dest by Jonathan_S   » Sun Jul 12, 2015 1:13 pm

Jonathan_S

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Joined: Fri Jun 24, 2011 1:01 pm
Location: Virginia, USA

Bill Woods wrote:
Jonathan_S wrote: Since RFC seems to ignore relativity for wedge acceleration, we can just use Newtonian formulas for all this.
I didn't, since 0.3c is mildly relativistic. Here's the set of formulas I've got:

[snipped the formulas]
Jonathan_S wrote: (Now in actuality I didn't work it out like this until now, I simply plugged the numbers into a spreadsheet I already had for missile accelerations; treating the ship like a very slow, very long endurance, missile)
Just what I did! If you compare, you'll see the author's error isn't plot-critical; missiles run slower, but they run longer so they actually have longer ranges than advertised.
Cool, thanks for showing the real math.

And it's kind of amusing that we both had spreadsheets for missiles to reuse for the ship accel.
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