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In calculating the specific impulse using the thrust generated as one of the parameters, do we take the total thrust generated or the average thrust over the period of the burn?

$I_{sp} = \frac{F_{Thrust} \cdot t_{burn}}{m_{fuel} \cdot g}$

If I am taking readings over a time interval of dt = 0.1 second for a total of 3 minutes, then do I take the thrust at dt=0.1 second or the average of the total thrust over a 3 minute period?

Also, during a launch, do we use the average thrust generated during the launch test to predict the projectile or do people have another way to find the thrust and make more precise calculation of the speed and position? As I know, the thrust is not steady during the launch.

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    $\begingroup$ "...using thrust generated as one of the parameters..." What are the other parameters you will use? The answer will depend on what other information you have available. And what does "total thrust" mean here? Do you instead mean "instantaneous thrust"? $\endgroup$
    – uhoh
    Oct 5, 2021 at 3:24
  • $\begingroup$ Hello @uhoh, I meant to ask when we say the thrust produced by a rocket is X, do we mean instantaneous thrust or the thrust in total over the period of the burn. $\endgroup$
    – Rocket Hack
    Oct 5, 2021 at 6:52
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    $\begingroup$ Both, really. one gives you instantaneous Isp, the other gives you average over the whole burn, which will tend to be less than peak instantaneous Isp. What is normally called "Isp" is the peak achievable instantaneous under optimal conditions, or under exactly specified conditions (with chamber pressure V, nozzle ratio X, at ambient air pressure Y, the Isp for this motor is Z) $\endgroup$
    – CuteKItty_pleaseStopBArking
    Oct 5, 2021 at 6:56
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    $\begingroup$ Look at the units. $F/(m_{\text{fuel}} g)$ is unitless, so that's wrong. $(\bar F t)/(m_{\text{fuel}} g)$ does have the right units (time), so that's okay with regard to average specific impulse. You should be using $\dot m_{\text{fuel}}$ and not multiplying by time for instantaneous specific impulse. $\endgroup$
    – David Hammen
    Oct 5, 2021 at 12:59
  • $\begingroup$ "...do we use the avg thrust generated during launch test to predict the projectile" - shouldn't you use acceleration instead? If you have engine gimbal you would need to take that in account as well. $\endgroup$
    – Bojan Kogoj
    Oct 5, 2021 at 13:27

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There's no "the" specific impulse, in the sense that the performance of a rocket engine varies over time. For a given moment in time, the instantaneous specific impulse can be derived from thrust and mass flow:

$$I_{sp} = \frac{F_{thrust}}{\dot{m}_{fuel} \cdot g}$$

For accurate modelling of a launch in a complex environment (say, in an atmosphere), we would need to know the engine's fluctuations to get exact results.

But having an $I_{sp}$ value for the whole burn makes just as much sense, as it lets you calculate change in velocity without having to worry about exactly how the burn progressed. In that case, average thrust would indeed be what you want:

$$I_{sp} = \frac{\bar{F}_{thrust} \cdot t_{burn}}{m_{fuel} \cdot g}$$

So both ways of calculating specific impulse are perfectly sensible, though for different use cases.

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    $\begingroup$ Nice answer, I almost missed the lack of a dot over the m in the 2nd equation. $\endgroup$
    – Organic Marble
    Oct 6, 2021 at 1:35
  • $\begingroup$ Thank you @SE . Using the average value for Isp and Thrust, would likely not give an precise info during the powered burn duration, but I assume, that would not make much of a difference in calculating things like final height and velocity at engine cut off. $\endgroup$
    – Rocket Hack
    Oct 8, 2021 at 10:15

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