Math going wrong. Can't calculate Delta-V using Newtons Law

[Thormack]

Hello guys,

I don't like to calculate Delta-V using ISP.

I prefer to use the good old Newtons Law F = M x A.

The problem is something is wrong with the spreadsheet I made. I'm posting a picture below, so you guys can analyze it and gime some feedback.

Thanks very much.

[Spricigo]

F=m*a is a simplification for when mass don't change.  That's not the case for rocket propulsion. 

 

 

 

[StahnAileron]

As @Spricigo notes, F = MA is for simple (i.e. static) calculations. DV calculations take time (i.e. varying mass over time) into account. You need calculus equations for that, not simple algebraic formulas. That's the whole point of ISP numbers: they encompass the calculus into a single value you can insert into simplified algebraic equations. I say just use ISP numbers. It's a discrete value rather than a derivative value for a reason.

[bewing]

Looks like you also forgot to include the mass of the oxidizer?

[Spricigo]

1 hour ago, bewing said:

Looks like you also forgot to include the mass of the oxidizer?

Seems to me it's not the case since he uses the mass of the tank to obtain the average mass (full,  empty)  the fuel amount(with fuel consumption)  is only used to obtain the burn time.

 But the equation for the average of full and empty mass is wrong. Considering the correct dry fraction, would be ((C14/9)+C14) /2 or even C14*5/9 instead of the actual formula the OP used.  

 

[GoSlash27]

Isp is actually just a measure of the fuel efficiency of the engine; if it has a mass of fuel equal to it's thrust it will run for (Isp) seconds. Multiply the Isp by g0 and you have the velocity of the exhaust. It's not a value that's derived from calculus.

 Part of the problem is that this calculation doesn't take into account the diminishing mass of the rocket, but the bigger problem is that it doesn't reflect what a rocket does.

In the simplest form it takes part of the mass of the rocket, expels it from the rear at high speed, and imparts an equal and opposite reaction in the rocket. m1v1=m2v2.

If you have the thrust of the rocket and fuel mass flow rate, then you have the exhaust velocity. Vx=T/m*.  Vx*ln(Mw/Md)=DV.

tl/dr... If you don't want to use 9.81*Isp in your calculation, just use T/m* instead. They're the same thing.

HTHs,
-Slashy

 

 

[Spricigo]

1 hour ago, GoSlash27 said:

Isp is actually just a measure of the fuel efficiency of the engine; if it has a mass of fuel equal to it's thrust it will run for (Isp) seconds. Multiply the Isp by g0 and you have the velocity of the exhaust. It's not a value that's derived from calculus.

Actually calculus is needed to derive Tsiolkovsky Rocket Equation from Newton's 2nd Law.  Apart from this nice reply.

[paul23]

Newton's law is NOT F = m*a

 

Newton stated:

The force on an object equals the rate of change in momentum.

 

 

The rocket equation is "easily" calculated from this, and even if you do it by hand you WILL have to do these calculations: there's no getting around it as it is just the solution to above equation (F=m*a is another solution in a situation where mass doesn't change with time).

 

Some trickery of variables (correct definitions of exhaust velocity):

Since external forces are zero, sum of all forces needs to be zero (conservation of momentum), and above can be rewritten:

This is a first orde ODE, which calculus teaches us to solve easily (coincidally newton explained this to us); 

 

 

So this is not just a "preferable" way to solve/calculate the change in velocity/acceleration: rather it is the ONLY correct way to calculate these things. Using the average mass is incorrect since the accelaration is way higher near the "empty" point.

https://en.wikipedia.org/wiki/Tsiolkovsky_rocket_equation#Derivation

 

 

 

And just to state the terms: the specific impulse (Isp) is indeed just a way to "measure things in normal values for comparison": though in other formulas Isp is useful above others.

The "V_e" used above is the so called "effective exhaust velocity": it would be the exhaust velocity if all energy is put into velocity change of the exhausting medium and no longer there is pressure; this implies that the exhaust has the same pressure as the atmosphere. Which obviously can't be the case since in vacuum we would need near zero Pascal pressure, while at atmospheric level the pressure is 100 KPa.

[GoSlash27]

4 hours ago, Spricigo said:

Actually calculus is needed to derive Tsiolkovsky Rocket Equation from Newton's 2nd Law.  Apart from this nice reply.

That it is. I'm just pointing out that it's not needed to derive Isp.

Best,
-Slashy

[FancyMouse]

And if it interests you, the same principle of energy and momentum conservation can be used to solve the relativistic rocket equation. There are some fun consequences, especially if you propel stuff at speed of c.

[wumpus]

12 hours ago, StahnAileron said:

As @Spricigo notes, F = MA is for simple (i.e. static) calculations. DV calculations take time (i.e. varying mass over time) into account. You need calculus equations for that, not simple algebraic formulas. That's the whole point of ISP numbers: they encompass the calculus into a single value you can insert into simplified algebraic equations. I say just use ISP numbers. It's a discrete value rather than a derivative value for a reason.

If you are already using a computer to compute these things, you can bypass the calculus bits by including the rates of change in your calculation:

if a=F/m, then compute the acceleration for each second, while reducing the mass according to how much the rocket engine burns per second.  The delta-v you calculate will be the sum of all the accelerations (until the fuel runs out).

Expect errors in this method, but you can improve the accuracy by reducing the time intervals you calculate a=F/m over (and of course, divide acceleration by the time interval and any other issues ignore due to unit measures).  While purists may scoff at such tricks, this is exactly how KSP calculates such things in game (at least in the "physics bubble").  In practice, just use the logarithm (or Kerbal Engineer).  But it certainly helps to learn where they came from (especially if you have to deal with complicated issues such as solid boosters burning out while liquid rockets keep firing).

[StahnAileron]

1 hour ago, wumpus said:

If you are already using a computer to compute these things, you can bypass the calculus bits by including the rates of change in your calculation:

if a=F/m, then compute the acceleration for each second, while reducing the mass according to how much the rocket engine burns per second.  The delta-v you calculate will be the sum of all the accelerations (until the fuel runs out).

Expect errors in this method, but you can improve the accuracy by reducing the time intervals you calculate a=F/m over (and of course, divide acceleration by the time interval and any other issues ignore due to unit measures).  While purists may scoff at such tricks, this is exactly how KSP calculates such things in game (at least in the "physics bubble").  In practice, just use the logarithm (or Kerbal Engineer).  But it certainly helps to learn where they came from (especially if you have to deal with complicated issues such as solid boosters burning out while liquid rockets keep firing).

That's not bypassing it; that's doing it. That IS the calculus part.

This is why simple DV calculations use ISP numbers: ISP calculations already take varying mass and acceleration over time into account (which is the calculus part needed). If you want to bypass calculus, you just use the ISP numbers.

Edited December 29, 2016 by StahnAileron

[wumpus]

27 minutes ago, StahnAileron said:

That's not bypassing it; that's doing it. That IS the calculus part.

This is why simple DV calculations use ISP numbers: ISP calculations already take varying mass and acceleration over time into account (which is the calculus part needed). If you want to bypass calculus, you just use the ISP numbers.

You want Delta-v numbers from Kerbal Engineer or mechjeb.  All the ISP numbers are is exhaust velocity divided by the acceleration of (a unit) gravity.  They are useful for being the same for metric and imperial units, little more.  Of course, exhaust velocity has its own nasty equation with various variables and constants hanging all over it, so expect to look up that (or ISP) in a table.

ISP has nothing to do with "bypassing calculus" (or the rocket equation).

[GoSlash27]

19 minutes ago, wumpus said:

ISP has nothing to do with "bypassing calculus" (or the rocket equation).

^ This, as I said earlier. The rocket equation itself is derived using calculus, but specific impulse is just straight algebra. It's simply exhaust velocity divided by standard gravity or thrust divided by mass flow rate.

 Best,
-Slashy

 

[Spricigo]

I think we are making complementar  arguments sound like counterpoints. 

Let's try to summarize:

StahnAileron, Paul23 and I  pointed :putting thrust and average mass in F=m*a will not give the average acceleration to be used to find total deltaV.  Also that starting from Newton's 2nd Law one need some calculus to get correct deltaV

GoSlash27 and wumpus  pointed that,  given the fact there is a algebraic equation for deltaV and also mods/addons that simple show deltaV, there is no need to dwell in calculus.  

FancyMouse poundered about using principle of energy and momentum conservation to solve relativistic rocket equation.  (//sound like an interesting discussion to me but maybe for another thread) 

IMO there is 2 confusing moments in the discussion:

1)  GoSlash27 said "Isp is not derived from calculus" ; my nitpick "TRE is derived from N2L"  GoSlash27 clarification "that's it.  Nothing to do with my point." (no response from me since he made clear both points are valid and independent) 

2)wumpus said "you can bypass the calculus considering deltaV=a.t with a=F/m and very small t so we can sum up all those small intervals while the rocket burn fuel" ; StahnAileron answered " what you described is calculus" 

IMHO we need to clarify (2).  Wumpus was just pointing we can use F=m*a and avoid symbolic calculus since we also have a computer to do the tedious number crunching . StahnAileron may be correct,  this is numeric calculus but wumpus point was "let the computer do the hard work" and "ISP is a shortcut that you will probably take" 

Excuse me if I misinterpreted something.  Felt like those considerations were needed to settle the question (to me at least). 

 

[GoSlash27]

10 minutes ago, Spricigo said:

"ISP is a shortcut that you will probably take" 

Spricigo,

 My main point would be that since there's nothing special about the Isp figure, there's no reason why the OP must use it in the rocket equation if he just doesn't want to. The rocket equation actually uses exhaust velocity, and he can pretty easily calculate that by his thrust and mass flow rate. It seems kinda silly to me to not just go ahead and use Isp since it's right there in the part description and .cfg file, but hey... to each his own *shrug*

Best,
-Slashy

 

[Spricigo]

1 hour ago, GoSlash27 said:

...  It seems kinda silly to me to not just go ahead and use Isp since it's right there in the part description and .cfg file, but hey... to each his own *shrug*

Agreed.

I also see no point to crunch the numbers myself except maybe for a few times to get the idea of how each factor affects deltaV. But after playing as industrial/trader in EVE Online (a.k.a. Spreadsheets Online)  for some years I have no right to talk against enjoying some game related maths. 

 

[Loren Pechtel]

8 hours ago, wumpus said:

If you are already using a computer to compute these things, you can bypass the calculus bits by including the rates of change in your calculation:

if a=F/m, then compute the acceleration for each second, while reducing the mass according to how much the rocket engine burns per second.  The delta-v you calculate will be the sum of all the accelerations (until the fuel runs out).

Expect errors in this method, but you can improve the accuracy by reducing the time intervals you calculate a=F/m over (and of course, divide acceleration by the time interval and any other issues ignore due to unit measures).  While purists may scoff at such tricks, this is exactly how KSP calculates such things in game (at least in the "physics bubble").  In practice, just use the logarithm (or Kerbal Engineer).  But it certainly helps to learn where they came from (especially if you have to deal with complicated issues such as solid boosters burning out while liquid rockets keep firing).

What you are describing is the essence of calculus.  Those equations we hate in calculus are simply a way of reducing the interval to zero and doing it all in one equation rather than step by step by step.

Note that in most cases you do not need to know calculus to deal with the equations that come from calculus.

Consider an object being accelerated at acceleration a.

Acceleration = a

Velocity = a * t

Distance = 1/2 a * t^2

We are just given those without a good picture of why.  However, once you know calculus you see that the second equation is simply the integral of the first and the third is the integral of the second.  Or you can express it as the second is the derivative of the third and the first is the derivative of the second.

Once you understand this it becomes obvious how to calculate velocity and distance given non-uniform acceleration--it's still simply an integration step.

[Thormack]

Thanks very much for comments guys.

Anyway, can anyone point more exactly what is so wrong with my spreadsheet?

As I'm getting around half the actual delta-v that I should as result.

I might have to do some mass adjusts, nevertheless, there must be something else wrong besides it.

 

 

[GoSlash27]

1 hour ago, Spricigo said:

Agreed.

I also see no point to crunch the numbers myself except maybe for a few times to get the idea of how each factor affects deltaV. But after playing as industrial/trader in EVE Online (a.k.a. Spreadsheets Online)  for some years I have no right to talk against enjoying some game related maths. 

 

 I actually *do* crunch the numbers myself using a spreadsheet and I'm a big advocate of that approach, I just do it backwards. It's how I zero in on cheap and efficient designs before I build.
 There's a lot to be said for doing the math yourself

-Slashy

Edited December 30, 2016 by GoSlash27

[GoSlash27]

2 hours ago, Thormack said:

Thanks very much for comments guys.

Anyway, can anyone point more exactly what is so wrong with my spreadsheet?

As I'm getting around half the actual delta-v that I should as result.

I might have to do some mass adjusts, nevertheless, there must be something else wrong besides it.

 

 

Thormack,

 They have stated very explicitly what is wrong with your spreadsheet: It doesn't account for the shrinking mass of your rocket as it burns fuel. That's not the only problem with it, but it's the major one.

Best,
-Slashy

[FancyMouse]

2 hours ago, Thormack said:

exactly what is so wrong with my spreadsheet?

You can't use "average mass" to get around the fact that the mass is changing over time. It doesn't work that way.

[Wcmille]

@Thormack if you are not comfortable with calculus, try this:

Instead of using average mass to do your calculations, use a row in your spreadsheet to calculate the change in speed and mass each second. After you've set up the first two rows, then just clone the rows until the rocket is out of fuel.

It won't work out *exactly* the same, because you are still making approximations, but it should be much closer.

[Spricigo]

3 hours ago, Thormack said:

 

 

6 minutes ago, Wcmille said:

@Thormack if you are not comfortable with calculus, try this:

Instead of using average mass to do your calculations, use a row in your spreadsheet to calculate the change in speed and mass each second. After you've set up the first two rows, then just clone the rows until the rocket is out of fuel.

It won't work out *exactly* the same, because you are still making approximations, but it should be much closer.

Also you will notice later intervals will add more deltaV than early ones 

Edited December 30, 2016 by Spricigo
Missquote

[paul23]

The rocket equation actually uses exhaust velocity, and he can pretty easily calculate that by his thrust and mass flow rate.

 

Does KSP give the instantaneous/constant mass flow rate anywhere?