# Difference between Optimum Expansion Ratio and Maximum Thrust

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.