# Deriving the thrust of a rocket

I understand the the thrust of a rocket is given by

$$F_x = M_e V_e - M_i V_i + (P_e - P_i) A_e$$

However when I try to derive it myself using Reynold's transport theorem I get the following:

Below is my control volume of the rocket thrust chamber

• $A_i$ is the injector face area
• $P_i$ is the fluid inlet pressure, in absolute pressure (assuming pressurized by a turbopump)
• $A_e$ is the nozzle exit area
• $P_e$ is the atmospheric pressure
• $R_x$ is the reaction force required to hold the thrust chamber in place

Using Reynold's transport theorem, I get:

$$ΣF_x = M_e V_e - M_i V_i$$

$$P_i A_i - P_e A_e + R_x = M_e V_e - M_i V_i$$

$$R_x = M_e V_e - M_i V_i - P_i A_i + P_e A_e$$

This is different from what is mentioned in the third line above.

• Interesting question. If you want try MathJax here's a good link: meta.math.stackexchange.com/questions/5020/… For example, if you use asterisk twice that causes italics font. Instead of Me Ve - Mi Vi you can have $M_e V_e - M_i V_i$ by typing "M_e V_e - M_i V_i" and putting a dollar sign before and after it. For simple multiplication it's better to skip the asterisk. I'll edit your first equation so you get an idea. By using double dollar signs, it centers the equation, while a pair of single dollar signs keeps the equations in-line.
– uhoh
Nov 26 '16 at 12:18
• It is not clear what exactly is your question. Perhaps you should elaborate about what responses you are looking to get in return.
– TRF
Dec 22 '16 at 22:09
• you have failed to account for the de laval nozzle, which accelerates the flow in a supersonic condition which decouples it from the injection pressure
– user20636
Jul 7 '18 at 7:18