# Apollo 11 liftoff acceleration function

1. Dec 17, 2013

### leehufford

Hello,

Recently I saw a youtube video of the Apollo 11 launch with time and velocity data. I took data in 5 second increments and put the data in my graphing calculator. I used a quadratic regression to come up with a velocity time function.

I then differentiated this function to come up with an acceleration time function, and for the first 22 seconds I have a negative acceleration. Where did I go wrong?

My velocity function from quadratic regression:

v(t) = 0.1368t^2 - 5.9846t + 102.863

Then differentiating for a(t):

a(t) = 0.2736t - 5.9846

I tried removing the point (0,0) so I could do an exponential regression, but this fits the data worse and gives a nonzero acceleration at t=0...

So either I am using the wrong regression type or the data was bad. Is the launch simply too complex to fit into a nice differentiable function like this? I just thought it would be cool to have a handle on the types of acceleration experienced during this historic achievement. If you'd like me to post the data I would be happy to. Thanks in advance,

Lee

Last edited: Dec 17, 2013
2. Dec 17, 2013

### A.T.

3. Dec 17, 2013

### leehufford

4. Dec 17, 2013

### A.T.

That intercept of 102.863 doesn't make any sense. v(0) must be 0.

5. Dec 17, 2013

### leehufford

I agree. Do you think the data is bad? I've tried exponential and power regressions to no success. Not sure how to force the regression to go through the point (0,0). I double checked my numbers and they agree with the numbers in the video. Do you think the velocity function should be quadratic? Thanks for your help.

-Lee

6. Dec 17, 2013

### rcgldr

I recall something like 7 million pounds of thrust on a 6 million pound Saturn V rocket full of fuel. The initial launch acceleration would be fairly slow, 1/6 g upwards.

7. Dec 17, 2013

### SteamKing

Staff Emeritus
You give formulas for v(t) and a(t), but the independent variable is 'x'. Shouldn't 'x' be 't'?

8. Dec 17, 2013

### leehufford

Yea I don't know why I mixed those up, I went ahead and edited that so t is the independant variable.

I wonder if the actual function itself was changing during the launch, making the concept of a single function invalid. My data went from t=0 until t=160 seconds because at that point the first stage rocket is jettisoned and there is an abrupt change in the velocity.

I wonder if shortening the timespan would improve the function. I know usually the more data the better, but if it does, it might indicate that the data should not be considered as one function, but several.

-Lee

9. Dec 17, 2013

### SteamKing

Staff Emeritus
During stage separations, the rockets in the jettisoned stage have been cut off and the rockets in the next stage have not ignited, so the rocket is not accelerating during these intervals, at least due to thrust. There is a change in mass though from the loss of the stage being jettisoned.

By not including any data points during the first stage burn, you have necessarily lost accuracy in crafting your velocity and acceleration functions.

10. Dec 18, 2013

### leehufford

I'm a little confused now. I thought the time interval from t=0 to t=160 was the first stage burn, which is the only time I considered. I excluded time after that because I thought the velocity change would mess up the regression for the first part of the launch. Thanks for the relpy.

-Lee

11. Dec 18, 2013

### harkkam

From what I understand, the equation that was created cannot be the right equation simply because of the y intercept first of all being more than 0.

So either the data was entered incorrectly or the regression function is not working properly. Perhaps run the regression in an excel spreadsheet vs a graphing calculator and try again.

I would think that a rocket is just a simple object accelerating upwards, like those ball toss problems that physics 101 text books use.

Here the ball is the rocket.

However there might be something to consider, since the rocket is consuming fuel very quickly the change in mass is taking place, its getting lighter and the accelerating forces on a changing mass might cause the function to approx incorrectly, as the force acted upon the rocket remains the same.

In other words I think since the rocket is producing I would assume consistent thrust per time unit, but accelerating a changing mass, that might be a problem.

12. Dec 18, 2013

### leehufford

That was essentially my idea behind shortening the timespan, which would reduce some of the mass-loss effect. Not sure about the ball/rocket comparision though... one has an upward thrust combatting gravity while the other is just combatting gravity right?

-Lee

13. Dec 18, 2013

### harkkam

Yes, the ball has a negative acceleration, i guess what I was trying to say was that my first thought was that this problem is a simple motion problem, negative and positive acceleration are in my opinion the same type of force but in different directions and you can "plug and chug".

But then I realized that the rockets acceleration is not constant unlike the ball, which is 9.8/ms^2,

so yeah they are different, but I think that the derivative of acceleration of the right eq, would give us how fast the acceleration changes, but still not sure how to help solve the problem.

14. Dec 18, 2013

### harkkam

perhaps, Im not sure but maybe a group of functions needs to be created, each unique function for maybe i dont 1 sec time intervals and plot each function on a graph and then approx one function from that, not sure if that makes sense, just something i came up with

15. Dec 18, 2013

### K^2

That seems very unlikely. TWR only slightly above 1 is extremely wasteful. Optimal for vertical ascent portion is 2, but since the whole thing gets lighter as it burns fuel, you'd typically start a touch low, and end up a touch high. Still, anything bellow 1.6-1.7 would be just a waste of fuel. You are almost better off to just pour out the fuel that's holding you back and start with higher acceleration.

Edit: And yet, that seems to be what they have gone with. I'm getting TWR of 1.23 from data on Wiki. I'll have to take a look at how that changes as fuel is burnt off, but this just doesn't seem like smart engineering, which really shouldn't be the case. Unless, increase in payload requirements came in late, and they had to increase the size of first stage tanks to compensate. But even then, some boosters would do a much better job of it... Weird.

Edit 2: Curiouser and curiouser. I just plugged in some numbers.

At the listed thrust of 34MN and ISP of 263s, in the 150s that the first stage was firing, it'd burn through just under 2,000T of fuel. Which is consistent with masses listed. Plugging that into the rocket formula, I get total $\Delta V$ of 3,160m/s. But hanging out in Earth gravity for 150 seconds brings it down to 1,690m/s.

Now, suppose I just poured out 600T of fuel. I really want that higher TWR on liftoff. No modifications to the rocket. We just reduce initial mass of the rocket, keeping final mass, after stage burns through, unchanged. Naturally, we get much less $\Delta V$. Just 2,540m/s, in fact. But we also burned through all this fuel in just under 105 seconds instead of 150. After we take gravity's toll out, we're down to 1,515m/s.

I don't know about you, but 175m/s for almost a third of your fuel seems a bit wasteful. And that's without taking into account any savings you can make by reducing tank size.

Now, without taking drag into effect, this does buy you about 20km of extra altitude. That's significant on the ascent stage. Difference between 60km and 80km when you fire second stage would be huge. Of course, rocket would be far into rotation by then. But I have a feeling that once I add even a simple atmospheric model, I'm going to lose even that advantage. I'm going to go check that now.

Edit 3: Alright, so even with atmosphere in place, it looks like this extra fuel is buying altitude. None of the above takes into account the fact that ISP of the engines improves dramatically up there, but the idea is the same either way. Get as much fuel to thin atmo as you can, and let it all burn there. The whole thing could still be more efficient with a bit more thrust at the liftoff, but I'm guessing it comes down to working with what you have. I'm glad I got that sorted. Maybe it will be interesting to someone else.

Last edited: Dec 18, 2013
16. Dec 18, 2013

### sophiecentaur

Rockets are hugely inefficient in the initial stages (zero efficiency for the first second or so). You just have to put up with virtually chucking away most of your fuel during and just after liftoff. Space engineers are constantly looking for better ways of launching but nothing viable has yet turned up. I am really surprised that aircraft launching is not yet the thing to use.

17. Dec 18, 2013

### SteamKing

Staff Emeritus
It shouldn't be that surprising that A/C launch is not used. Rockets are still pretty sizable objects to fly around with. Granted, later systems use solid propellant instead of liquid fuel, but I think the desire is to increase the amount of payload which can be put into orbit. Even a Saturn V type rocket with the first stage omitted is a big critter.

http://en.wikipedia.org/wiki/Delta_IV

http://en.wikipedia.org/wiki/Antonov_An-225_Mriya

From the articles, you can see that the Delta IV launch system weight can easily exceed the maximum payload capacity of the An-225 aircraft.

Now the Russians have lately become interested in using the An-225 in an air launch project:

http://en.wikipedia.org/wiki/Air_launch_to_orbit

Who knows what will happen?

18. Dec 18, 2013

### cjl

Another thing that the lower TWR buys you is lower dynamic pressure. If you look at your velocity vs altitude profile, you'll find that the one with the higher TWR at liftoff will have a significantly higher velocity through the lower atmosphere, which increases aerodynamic loads on the vehicle structure. You can somewhat compensate for this by throttling down shortly after liftoff, and then throttling back up after you are through the thickest part of the atmosphere, but that increases gravity loss. You also tend to want to throttle down near the end of the stage burn to limit peak acceleration, and at this point, since you're throttling down for dynamic pressure at the beginning of the burn, and throttling down to limit acceleration at the end of the burn, you really start to get to the point where it's worth downsizing the engines and living with the low TWR at liftoff.

Also, another thing that you might not be considering here is the relative cost - rocket engines are very expensive, but rocket fuel is (relative to the cost of a launch) cheap. The fuel cost is a tiny percentage of the overall cost of a modern launch vehicle, so if you can gain any benefit by increasing fuel load without increasing engine count or thrust, even if it is hugely wasteful, it is probably worthwhile. This is especially true with liquid fuel rockets - if you look at the liftoff TWR of rockets powered only by liquid fuel, they tend to be pretty slow off the pad, while ones with solid rocket boosters tend to jump off the pad much faster (though still not typically above 1.5 or 1.6:1 or so)

19. Dec 18, 2013

### cjl

As for the original question? You're taking 160 seconds of data from something that really isn't quadratic and trying to fit a quadratic to it - it isn't that surprising that your regression function doesn't match the data terribly well.

Last edited: Dec 18, 2013
20. Dec 18, 2013

### sophiecentaur

It surprises me that a launch system is pursued that involves such vast single payloads when there are other alternative philosophies which could be more economical. There is no problem in assembling large vehicles from smaller parts, once in orbit. (Ask IKEA) I know that Spaceships One and Two are a lighthearted approach to proper spaceflight but it is an attractive idea.
As you say "Who knows what will happen?" There could be a real rush to put more stuff up there, once launching gets cheaper.
I wonder if one reason for keeping the old system is security based. The major powers are wetting themselves at the thought of ballistic missiles for all and the very size of what's needed at present is a factor that keeps the space club membership limited.

21. Dec 18, 2013

### sophiecentaur

@cjl
Some really interesting points there. Every business has its own set of factors which are fascinating to the outsider.

22. Dec 18, 2013

### SteamKing

Staff Emeritus
The problem with assembling large payloads in space is that they don't assemble themselves. Somebody's got to be there to put all the parts together. Just look at how long it took to design and assemble the modest Space Station which is currently in orbit. If anything happens to the Russian space program, even this small accomplishment is placed in jeopardy.

23. Dec 18, 2013

### sophiecentaur

Yeah. I know that nothing's easy but, now and again, technology makes a leap. In this case, the leap could be robotics. You could imagine a robot factory up there with the little chaps standing there, waiting for a new kit to put together. Minimal 'life support' needed when there's nothing to do. My personal view is that most of that Space Stuff should be done without humans. The Wild Frontier thing is out of place most of the time. Just think of how much Curiosity has managed to do by not needing to support and protect a human research team.
Is 'being there' really so important?

24. Dec 18, 2013

### cjl

Generally (though admittedly with a couple exceptions), a single large launch vehicle is actually cheaper than launching an equivalent payload with multiple smaller vehicles, even ignoring the logistics and overhead of in-space assembly.

As for spaceship one (and two), I don't really see them going anywhere as far as real space exploration/commercialization/travel is concerned. The energy difference between suborbital and orbital flight is enormous, and as a result, what works well for one is not necessarily good for the other.

25. Dec 18, 2013

### jhae2.718

Look at Orbital's Pegasus launch vehicle.