Tip nozzle paradox

Whilst mulling over the mechanics of rocket tipped rotors I came up with 
the following paradox.

Consider a low ISP rocket, say 20 seconds; mounted on an arm pivoting on 
a central spindle. The propellent enters the rocket along a tube up the 
central spindle, goes through a frictionless coupling and then follows 
the tube along to the rocket tip.

The rocket is oriented so that the exhaust points at 90 degrees to the 
rotation axis.

Now from momentum considerations you can show that the tip speed should 
be equal to the exhaust velocity- in this case ~200 m/s (fuel has to be 
accelerated up to the tip and then leaves it at ~200m/s. Clearly the 
momentum balances when the tip goes at 200m/s.)

However this implies that the exhaust leaves the nozzle and stops; and 
hence has no energy, and hardly any heat, (rockets are typically 90% 
efficient at turning hot gas into moving gas, so the exhaust gas is 
relatively cool).

And yet the rocket clearly isn't accelerating and we have burnt all this 
fuel, which has liberated energy. Conservation of energy is the law!

Assuming there is no air drag or other friction; where has the fuel 
energy gone?

Winner gets 1kg of wishalloy.

"Lisa, in this house we *obey* the laws of thermodynamics" - H. Simpson