Does the speed in which a gas through a medium matter? There are two thrusters. They both expel the same volume of gas. The difference is, one has a fine stream of propellant at a higher velocity, and the other a wider stream at a lower velocity. Would both provide the same amount of thrust in Earth's atmosphere? or space?
 A: From Newton's second law, the rocket thrust force $$\tag 1F =\frac{\Delta p}{\Delta t} = \dot m v_e+m\dot v_e=\dot m v_e$$ where we assume a constant gas exit velocity or $\dot v_e=0$  and $$\tag 2\dot m=\frac{\Delta m}{\Delta t}=\rho v_e A$$ is the mass flow rate, $p$ is the momentum of the expelled gas, $\rho$ is the density of the expelled gas, $A$ is the gas exit area and $v_e$ is the exit velocity of the gas.
This means that $$F=\rho A v_e^2\tag3$$
We see that if you decrease the gas exit velocity and increase the exit area, and then if you increase the gas exit velocity and decrease the exit area, the gas exit flow rate in equation (2) will be equal in both cases, but the amount of thrust$^1$ given by equation (3) will in general not be the same due to the non-linear dependence of the exit velocity $v_e$
This thrust is independent on the pressure outside of the rocket, so ideally it will not depend all too much on whether or not the thrust is measured in the atmosphere or in space.
$^1$Note that this assumes ideal conditions, and that in each case, the density of the exiting gas stays the same. In real situations though, since parameters like temperature and pressure will vary in each case, the mass flow rates will not usually be the same.
Also note that propulsive efficiency varies slightly with different altitudes due to changing atmospheric pressure. One could conclude from this that the thrust will be slightly different in the atmosphere than in space.
