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when I look at the equation for thrust of a rocket: $$ F=\dot{m}*v_{exit}+(p_{exit}-p_{amb})*A_{exit} $$

In the book "Rocket Propulsion" from Sutton it says on page 33:

In the vacuum of space $p_{exit}=0$ and the pressure thrust becomes a maximum.

Further: When the exit gas pressure is less than the surrounding fluid pressure, the pressure thrust is negative. Because this condition gives a lower thrust and is undesirable for other reasons, rocket nozzles are usually designed that $p_{exit}=p_{amb}$ --> optimum expansion ratio.

Question: Why do I want to operate at optimum expansion ratio? Is this the point, where the thrust (apart of vacuum condition) is at its maximum for a specific ambient condition?

What I know: Normally according to the equation above, $p_{exit}$ should always be greater than $p_{amb}$ to maximize the thrust. But one forgets that $v_{exit}$ does not have its optimum when $p_{exit} > p_{amb}$.

Question 2: If I have designed a nozzle for let us say 10 000 meters and I know all my parameters at exit and throat and the expansion ratio for this nozzle "layout". Now I freeze this nozzle design. In order to calculate the thrust at 50 000 meters, is it enough to use $$ F=\dot{m}*v_{exit}+(p_{exit}-p_{amb})*A_{exit} $$ and use the layout values for the designed nozzle except for $p_{amb}$ ? Or will my $v_{exit}$ and $\dot{m}$ and ${exit}$ change with higher altitude?

What I know but what does not help me here: Thrust will increase with higher altitude.

Thank you for your precise answers!

Best regards Lucas

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  1. Nozzles for boosters are not "usually designed so that Pexit = Pamb" Ambient pressure changes during ascent, and tradeoffs are done during design. Vacuum nozzles are not designed so that Pexit = 0; that takes an infinitely long nozzle.

  2. Yes. To a first approximation, mass flow, exit plane pressure and exit velocity only change with throttle setting so the momentum thrust is constant and the pressure thrust varies with altitude. See https://space.stackexchange.com/a/46548/6944 for a worked-out example.

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