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# Determine Altitude in Kerbin Atmosphere of Stage

## Question

Hey everyone, I recently have been sucked into this game, and I'm loving the math. My question is this simple, determine the altitude a rocket will achieve on full fuel burn of a single stage. I've done a lot of research and have come up with the following example problem to test my algorithm/process of calculation. Let me know what you guys think of below and what I'm missing or potentially a force I haven't considered into the calculation such as lift, as you'll see my answer is off by nearly 3,300m. (For the sake of simplicity the rocket travels straight up in a vertical dimension only.)

Known Values of my Rocket:

• Full Mass     [MFull]       (Entire Rocket)              : 7.5t (7,500kg)
• Empty Mass [MEmpty]   (First Stage Depleted)   : 4.5t (4,500kg)
• Fuel Mass    [MFuel]      (Both LQ and OX)         : 3.0t (3,000kg)
• Isp                [Isp]          (Reliant Engine)            : 265 sec
• Thrust           [FT]          (Thrust Force Atm.)      : 205.2kN (205,200N)
• LQ Rate        [BLQ]        (Burn Rate of LQ)         : 7.105 u/sec
• OX Rate        [BOX]       (Burn Rate of OX)         : 8.684 u/sec
• LQ Volume    [VLQ]       (LQ Fuel 45% Mix)        : 270 u
• OX Volume    [VOX]       (OX Fuel 55% Mix)       : 330 u

Known Values of Kerbin:

• Accel. Kerbin [g]           (Accel. of Gravity)         : 9.81 m/sec^2

First I will calculate the time required to burn through the fuel mixture. This time will be needed in the final calculation.

Tburn =VLQ /BLQ =VOX / BOX << >> 270u / (7.105u/sec) = 38.0 sec

Next I convert burn rate units from volume/sec to units of kg/sec. (I assume 1u = 5kg of both LQ and OX)

BLQ_M = BLQ * (5kg/u) << >> (7.105u/sec) * (5kg/u) = 35.525 kg/sec

BOX_M = BOX * (5kg/u) << >> (8.684u/sec) * (5kg/u) = 43.42 kg/sec

BTOTAL= BLQ + BOX << >> 35.525kg/sec + 43.42kg/sec = 78.945 kg/sec   (M *Dot = Mass Flow Rate)

Determine effective exhaust velocity of rocket motor related to Specific Impulse and Gravity. (NASA Formula)

Ve= Isp * g << >> 265sec * 9.81m/sec^2 = 2,599.65 m/s

Determine acceleration of rocket (Found this formula on a physics forum, not sure if valid)

a = Ve ( BTOTAL / MFULL ) - g << >> 2,599.65 m/s * (78.945kg/sec  /  7,500kg) - 9.81m/s^2 = 17.554 m/s^2

Apply classical kinematic physics equation for displacement with acceleration. (Vertical Axis only...)

deltaX = 0.5 * a * (Tburn^2) << >> 0.5 * 17.554m/s^2 * (38sec ^ 2) = 12,673.988m

So in the end this calculation results in an effective altitude of 12,673.98 meters. If anything, I expect drag (if simulated) among other forces to take away from this value. Instead the opposite occurred, my actual test flight while holding steady to the center of the NavBall resulted in roughly 16,000 meters altitude at 38 seconds into flight (after stage finished burning).

Any ideas?

## Recommended Posts

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25 minutes ago, BattleReadyKen said:

Isp                [Isp]          (Reliant Engine)            : 265 sec

It's atmospheric ISP of Reliant, which means it only applies on surface level. Isp of the engine will increase quite fast during the ascent, so the final altitude will be higher.

Also drag is relatively small effect for an aerodynamic rocket.

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20 minutes ago, Abastro said:

It's atmospheric ISP of Reliant, which means it only applies on surface level. Isp of the engine will increase quite fast during the ascent, so the final altitude will be higher.

Also drag is relatively small effect for an aerodynamic rocket.

Ah that makes sense. NASA has it here that the thrust force, which factors into Isp, is based on pressure of the system. Thank you. This algorithm is still useful and I will probably compile it into a C++ executable as a side tool for myself in the future. If anyone stumbles upon this and would like a copy of the tool just let me know! I find this particularly useful for determining the minimum parts/mass required in the first initial stage(s) to leave Kerbin's atmosphere.

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54 minutes ago, BattleReadyKen said:

Ah that makes sense. NASA has it here that the thrust force, which factors into Isp, is based on pressure of the system. Thank you. This algorithm is still useful and I will probably compile it into a C++ executable as a side tool for myself in the future. If anyone stumbles upon this and would like a copy of the tool just let me know! I find this particularly useful for determining the minimum parts/mass required in the first initial stage(s) to leave Kerbin's atmosphere.

Calculating optimal first stage? I'm looking forward to it!

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Welcome to the forums, @BattleReadyKen - but please keep in mind that a vertical flight is not a useful test case for rocket stages. Altitude is only a minor factor; the vast majority of fuel is spent accelerating sideways. Knowing how much fuel you need to lift a payload out of the atmosphere isn't very helpful if the second stage decouples with 0 m/s horizontal velocity out of 2300 required to stay in orbit.

A typical first stage might pack around 1500-2000 m/s worth of dV, and decouple while still in the atmosphere but going sideways at a good clip.

Edited by Streetwind
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17 minutes ago, Streetwind said:

A typical first stage might pack around 1500-2000 m/s worth of dV, and decouple while still in the atmosphere but going sideways at a good clip.

Ah yes, you are correct. That is one of the first major issues I was wrestling with, once I elevate myself above 70kM, the horizontal velocity is negligible. My first attempts at the game were to burn straight vertical with a payload orbital burn stage capable of > 2400 m/s dV, however this is significantly much less efficient and almost double what may be required at an acute tilt around 8000m (at least that's what I've heard is efficient). Assuming I launch my rocket (a much larger- multi-stage one), at a slight angle to distribute velocity in both horizontal and vertical directions, I imagine actually calculating the resultant horizontal and vertical velocity after the stage(s) fire would be quite difficult. Or maybe it just involves a basic trig manipulation?

*Also, the formula a = V* ( BTOTAL / MFULL ) - g, 'g' would likely have to be used with a trig function with the angle of incline as the center of gravity is no longer exerting a force directly through the center of thrust.

P.S. I still do full vertical payloads then massive burns at AP like an idiot sometimes hahaha! I like the general 1500-2000 dV rule of thumb, but would like more exact calculations if it is feasible to do so. Thanks.

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I could see it being useful for the test contracts that want something high and slow, but that's about it.

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9 hours ago, BattleReadyKen said:

Determine acceleration of rocket (Found this formula on a physics forum, not sure if valid)

a = Ve ( BTOTAL / MFULL ) - g << >> 2,599.65 m/s * (78.945kg/sec  /  7,500kg) - 9.81m/s^2 = 17.554 m/s^2

Apply classical kinematic physics equation for displacement with acceleration. (Vertical Axis only...)

deltaX = 0.5 * a * (Tburn^2) << >> 0.5 * 17.554m/s^2 * (38sec ^ 2) = 12,673.988m

The acceleration will change with fuel burnt. Also the formula is indeed invalid because it comes from a assumption of constant mass.

As other pointed this is of little practical use, but if you want to know it take a look how or rocket modelists do it: http://www.rocketmime.com/rockets/rckt_eqn.html

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5 hours ago, Kryxal said:

I could see it being useful for the test contracts that want something high and slow, but that's about it.

Yea, any objective that is non-orbital would be a good application for early game contracts like those. That wasn't my main goal, but it's definitely a use.

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4 hours ago, Spricigo said:

The acceleration will change with fuel burnt. Also the formula is indeed invalid because it comes from a assumption of constant mass.

As other pointed this is of little practical use, but if you want to know it take a look how or rocket modelists do it: http://www.rocketmime.com/rockets/rckt_eqn.html

Neat link. In their example, they find the average mass of the rocket during flight and use it as a constant. However a lot of the aero terms will be hard to determine using KSP.

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The complexity of modeling the in-atmosphere behavior is substantial enough that it is very common to skip the calculation and simply design the rocket for 3,500 m/s of delta-v to low-Kerbal orbit. Once you're in LKO, then the delta- v requirements can be easily and accurately calculated.

Edited by FloppyRocket
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The AeroGUI can tell you many of the actual values for the aero terms.

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2 hours ago, BattleReadyKen said:

Neat link. In their example, they find the average mass of the rocket during flight and use it as a constant. However a lot of the aero terms will be hard to determine using KSP.

There is lots of assumptions and aproximations. That is because usually there is no pratical way to know for sure(too many variables), so people use that assumptions and aproximations to get a rough idea and hope for the best.

For a more precise result you will need to delve deeper in maths, probably use numerical analysis or/and calculus. Wich for KSP is usually too much trouble because you can just try and revert. And when you get a particular design that works the next try will likely be to improve it in some way.

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