Rocket Thrust Equations (2025)

On this slide, we have collected all of the equationsnecessary to calculate the thrust of a rocket engine.In arocket engine,stored fuel and stored oxidizerare ignited in a combustion chamber.The combustion produces great amounts of exhaust gas at hightemperatureandpressure.The hot exhaust is passed through a nozzlewhich accelerates the flow.Thrustis produced according to Newton'sthird lawof motion.

The amount of thrust produced by the rocket dependson the mass flow rate through the engine, the exitvelocity of the exhaust, and the pressure at the nozzleexit. All of these variables dependon the design of the nozzle.The smallest cross-sectional area of the nozzle is called thethroat of the nozzle. The hot exhaust flow ischokedat the throat, which means that theMach numberis equal to 1.0 in the throat and themass flow ratem dotis determined by the throat area.

mdot = (A* * pt/sqrt[Tt]) * sqrt(gam/R) * [(gam + 1)/2]^-[(gam + 1)/(gam - 1)/2]

where A* is the area of the throat, pt is the totalpressure in the combustion chamber, Tt is the total temperaturein the combustion chamber, gam is the ratio ofspecific heats of the exhaust, andR is thegas constant.

Thearea ratiofrom the throatto the exit Ae sets theexit Mach number:

A/A* = {[(gam+1)/2]^-[(gam+1)/(gam-1)/2]} / Me * [1 + Me^2 * (gam-1)/2]^[(gam+1)/(gam-1)/2]

Solving for the exit Mach number when we know the exit area ratio is quite difficult.But, we can use a computer program to iteratively solve the equation. Here'sa JavaScript program that solves for the Mach number when you specify the area ratio:

By default, the program Input Variable is theMach numberof the flow. Since the area ratio depends only on the Mach number andratio of specific heats, the program can calculate the value of thearea ratio and display the results on the right side of the outputvariables. You can also have the program solve for the Mach numberthat produces a desired value of area ratio.Using the choice button labeled Input Variable,select "Area Ratio - A/A*".Next to the selection, you then type in a value for A/A*.When you hit the red COMPUTE button,the output values change. The area ratio is double valued;for the same area ratio, there is a subsonicand a supersonic solution. The choice button at the right top selectsthe solution that is presented.

If you are an experienced user of this calculator, you can use asleek versionof the program which loads faster on your computer and does not include these instructions.You can also download your own copy of the program to run off-line by clicking on this button:

We can determinethe exit pressure pe and exit temperature Te from theisentropic relationsat the nozzle exit: pe / pt = [1 + Me^2 * (gam-1)/2]^-[gam/(gam-1)]

Te / Tt = [1 + Me^2 * (gam-1)/2]^-1

Knowing Te we can use the equation for thespeed of soundand the definition of theMach numberto calculate the exit velocity Ve:

Ve = Me * sqrt (gam * R * Te)

We now have all the information necessary to determinethe thrust of a rocket.The exit pressure isonly equal to free stream pressure at some design condition.We must, therefore, use the longer version of the generalizedthrust equationto describe the thrust of the system.If the free stream pressure is given by p0, therocket thrust equationis given by:

F = m dot * Ve + (pe - p0) * Ae

You can explore the design and operation of a rocket nozzle withour interactive nozzle simulatorprogram which runs on your browser.

The thrust equation shown above works for both liquid rocket and solid rocket engines.There is also an efficiency parameter called thespecific impulsewhich works for both types of rockets and greatly simplifiesthe performance analysis for rockets.

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