The Orion Project --- do the predicted speeds violate the Tsiolkovsky Equation? I understand that a conventional rocket may move faster than its thrust but IIRC the equation developed in the 19th century the speed is limited to 4/3 the thrust speed.
Now, I understand that the Orion propulsion is not a rocket but it still has thrust and this thrust I am certain is nothing remotely close to moving at one percent of the speed of light.
So how does this work? Is the pusher-plate approach somehow radically different than a conventional rocket and if so, would dropping conventional explosives behind a spacecraft also be better than using a rocket? My strong intuition says no, so why did Orion's developers expect not just .01 c but I think even as high as 1 percent or maybe I saw, unbelievably, 10 percent of the speed of light. In fact, I am not sure why any speed was not possible.
 A: You write:

IIRC the equation developed in the 19th century the speed is limited to 4/3 the thrust speed.

But this isn't a rule I've heard of! You are right that the result is very sensitive to the exhaust velocity, but the Newtonian Tsiolkovsky rocket equation says that the final rocket velocity is the exhaust velocity multiplied by the logarithm of the final mass over the fueled mass, which in theory and in practice can be much larger than 4/3.
Using information from the Wikipedia page on the Tsiolkovsky rocket equation, we can start plugging in values for our change in rocket velocity $\Delta v$ given exhaust velocity $v_e$ and mass ratio $m_0/m_1=\text{fueled mass} / \text{dry mass}$:
$\Delta v=c \tanh(v_e/c \log(m_0/m_1))$. For values similar to those on the Wikipedia page, we can get velocities a significant fraction of the speed of light.

For the mass ratio $m_1/m_0$, I am using a range of $2$ to $10$. For the exhaust velocity, I use a range of 0 to 30000km/s (10% the speed of light). This extremely high exhaust velocity is based on this quote from Wikipedia:

[Freeman Dyson's] 1968 paper "Interstellar Transport" (Physics Today, October 1968, pp. 41–45) retained the concept of large nuclear explosions but Dyson moved away from the use of fission bombs and considered the use of one megaton deuterium fusion explosions instead. His conclusions were simple: the debris velocity of fusion explosions was probably in the 3000–30,000 km/s range and the reflecting geometry of Orion's hemispherical pusher plate would reduce that range to 750–15,000 km/s.

If the exhaust velocity is limited to a much lower value (30 km/s), we get a much lower graph of possible final velocities:


Mathematica code to generate the first graph:
ContourPlot[Tanh[3.3356409519815205`*^-6 exhaust Log[massRatio]],{exhaust,1,30000},{massRatio,2,10},FrameLabel->{"Subscript[V, e] (km/s)","Subscript[m, 0]/Subscript[m, 1]"},ContourLabels->(Text[ToString[#3]<>"c",{#1,#2},Background->White]&),PlotLabel->"Tsiolkovsky rocket equation\n\[CapitalDelta]V=c Tanh[Subscript[v, e]/c Log[Subscript[m, 0]/Subscript[m, 1]]"]
Second graph:
ContourPlot[QuantityMagnitude[1.0 c,"km/s"]Tanh[3.3356409519815205`*^-6 exhaust Log[massRatio]],{exhaust,1,30},{massRatio,2,10},FrameLabel->{"Subscript[V, e] (km/s)","Subscript[m, 0]/Subscript[m, 1]"},ContourLabels->(Text[#3 "km/s",{#1,#2},Background->White]&),PlotLabel->"Tsiolkovsky rocket equation\n\[CapitalDelta]V=c Tanh[Subscript[v, e]/c Log[Subscript[m, 0]/Subscript[m, 1]]"]
