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Can someone check my math?

Posted by: quantumg - Wed Jun 27, 2007 1:51 am
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Can someone check my math? 
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Post Can someone check my math?   Posted on: Wed Jun 27, 2007 1:51 am
The Wikipedia article for Armadillo Aerospace lists some specs for Pixel and Texel:

Gross Lift Off Weight (GLOW): ~1500 pounds
Thrust: ~3000 lbf

Which makes sense, if you want to hover 1500 pounds, you need at least 1500 lbf. I also says:

Burn time: >180 seconds

Which has been demonstrated. So is that burn time for full thrust?

Let's assume it is. I believe John Carmack has said in another post that they don't intend to go faster that 400 km/hr, which is about 111 m/s. Half the thrust will always be going to beat gravity, so we can estimate the maximum acceleration of the vehicle to be 9.8 m/s/s, so it takes only 11.3 seconds to get to 400 km/hr. This leaves us 168.7 seconds of thrust, but as we'll no longer be accelerating, we'll only be using half thrust from now on, so we can double this to 337.4 seconds.

In the 11.3 seconds used to accelerate to 111 m/s, the rocket has travelled 625.681 meters (0.5 * 9.8 * 11.3 * 11.3). In 337.4 seconds at 111 m/s it will travel 37451.4 meters. This gives the rocket a total height of 38077.081 meters.

Or 38% of the way to space. :)

Of course, you might want to keep some fuel to land it.

Also, the rocket is going to be expending fuel during these burns. For the short burn it is probably negligable, but for the long burn it could have a significant effect. So we need this other spec:

Dry weight: 650 pounds

Which tells us that the fuel weighs 850 lbs. If we calculate the average weight of the fuel during the long burn, it's probably something like 400 lbs. This means the acceleration of the vehicle is now much higher as we can think of the vehicle only weighing 1100 pounds instead of 1500 pounds.

As we only (on average) require 1100 lbf to keep the rocket slowing down due to gravity, we can more than double the 168.7 seconds of remaining burn. We can multiply it by 2.727 (3000 / 1100) to get 460.04 seconds of burn. At 111 m/s it will travel 51064.98 meters. This gives the rocket a total height of 51690.6649 meters.

Over half way to space!


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Post Re:Can someone check my math?   Posted on: Wed Jun 27, 2007 11:03 am
quantumg wrote:
Which has been demonstrated. So is that burn time for full thrust?

Unfortunately that appears to have been be the peak available thrust. The number of pound-seconds of impulse available is nearer a third of the value you use. Easy to estimate from the geometric mean of the start and end weight multiplied by the hover time. They can probably extend the hover time to about 210 seconds and with the engine working at altitude that might increase by a few tens of seconds. Consuming the propellant faster also increases ISP somewhat.
quantumg wrote:
I believe John Carmack has said in another post that they don't intend to go faster that 400 km/hr, which is about 111 m/s

In fights to high alitude, John intends to keep the density corrected airspeed at that level, so as altitude increases and air density reduces the true speed will be increasing to maintain that drag level. John has estimated an approximately 100 second burntime, the thrust starting off at maxium then backing off, before throttling back up again. Optimal thrust profiles share that basic pattern.

Fortunately while you do need to consider a very large number of vehicle states, it turns out not to be necessary to compute a near infinite number of thrust-profiles in order to find the optimal one, once you know the drag coefficient as a function of Mach number, engine charachteristics etc. It's interesting that it's broadly the same than those I found manually (so no very large improvements proved possible)- but sometimes there's a new feature near the end which I hadn't considered.

I expect that Pixel, without aerodynamic improvements might be capable of reaching approximately 12km, with them it might reach 20km or more. Essentially the same perfomance in a very long, skinny vehicle would be sufficient to reach 100km.


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Post    Posted on: Tue Apr 08, 2008 11:25 am
It is also important to remember that rocket doesn't stop moving up when the burn stops. After the burn you just no longer have any "positive" acceleration, and you are decelerated by drag and gravity until your vertical velocity reaches zero, which is when you are apogee.

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Post    Posted on: Mon Jun 09, 2008 5:08 am
Calculus is required for total speed due to increased vehicle acceleration while burning fuel...and estimating effects of drag due to airspeed - you can pretty well ignore that under about 50mph or over 20 miles up but it'll kill you otherwise 8)


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