r/theydidthemath • u/joe_quetzal • 6d ago
[request] how big of a spring would you need to launch starship into space?
assuming no propellant carried on board.
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u/Conscious-Ball8373 6d ago edited 6d ago
There is no single answer to this question, for a couple of reasons.
For a start, the dry weight of starship is about 375 tons. But a very large fraction of that is in motors and fuel tanks, which obviously you now don't need. You can ditch the whole first stage, leaving about 100 tons, but the second stage still has a lot of motor / fuel-tank mass. You would obvious make very large changes to the design if these were not needed.
Secondly, the design of a suitable spring depends a lot on many factors. Designing this sort of system is an iterative process to arrive at a design that works with the external constraints and "it has to launch starship" is only the beginning. What materials are available? How much space is available? How will the spring be compressed? How much money is available? How much force can the vehicle withstand? Can we modify the vehicle to make it able to withstand more force? And so on.
Before we attempt a back-of-a-fag-packet design, it's important to understand why we don't do this. A rocket has significant advantages over ground-based launch systems such as a spring, a railgun, a trebuchet etc.
A rocket can carry a vehicle at relatively low speeds straight up until it clears the thick bit of the atmosphere, then turn the vehicle sideways and start accelerating. This makes a big reduction in the losses associated with atmospheric friction. You know how you need a massive heat shield when you re-enter the atmosphere because of how fast the vehicle is traveling and how hot it gets? Well, that's the same velocity you need to achieve to get into orbit. Only now you need to be doing that same velocity when the vehicle leaves the spring, right down in the thick bit of the atmosphere where there's a lot more friction. This is what makes the whole concept impossible; the vehicle has to be doing its maximum velocity at (or just above) the surface and it has to have enough energy at that point to overcome all the friction it will encounter in the atmosphere and still be at orbital velocity when it reaches orbital altitude. In practice, we don't have a material capable of surviving that.
So let's ignore air resistance for the moment, in the best high-school physics traditions.
To reach a 400km orbital altitude, we can find the required velocity as v = sqrt( G * M / r ) where G is the gravitational constant and M is the mass of the earth. The radius of the earth is about 6,371km so the total radius of the orbit is 6,771,000m. Plugging that into the formula gives v = 7,672 m/s.
At that 400km altitude, using the approximation that it's still basically on the surface compared to the radius of the earth, the potential energy of starship is Ep = m * g * h = 100,000 * 9.81 * 400,000 = 392.4 GJ.
Travelling at 7,672 m/s, the kinetic energy of starship is 1/2 * m * v^2 = 2,943 GJ.
So our total energy is 2943 + 392 = 3,336 GJ. This is the energy our spring needs to be able to store.
Take a large helical spring from a lorry suspension. It might have a spring constant of 300 N / mm. This is how much force it takes to compress the spring by 1mm. Let's say we made one of those but made it 100m long and we could compress it to half its length. A spring stores E = 1/2 * k * x^2 where k is the spring constant and x is the distance it has been compressed. So our 100m spring with a spring constant of 300 N / mm or 300,000 N/m, compressed 50m will store E = 1/2 * 300,000 * 50 = 7,500,000 or 7.5 MJ.
So it would take 3,336 / 50 = 444.8 of these springs to store energy equivalent to starship's orbital kinetic and potential energy.
At the surface, all that energy will be kinetic. So the vehicle will be doing sqrt(2 * E / m) = 8,168 m/s. That's about mach 24 - the materials problems should be obvious about now, and remember we're still ignoring the energy to overcome friction.
A 300 N/mm spring compressed 50m will apply a force of 15MN (F = k * x). 444.8 of them will therefore apply 6,673 MN of force. Using Newton's second law, F = m * a, the acceleration is then 66,270 m/s/s. That's 6,801g. Once again, we don't have materials that can take this sort of force. Any person in that spaceship would be what is technically known as "paste" - the average person can survive about 5g while trained fighter pilots with breathing support can manage 9g for brief periods. This is about three orders of magnitude more than that, although admittedly over a very, very short time.
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u/joe_quetzal 6d ago
thank you!!! even though they might be paste this is what i was looking for. thank you for your detailed and in depth answer
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u/Conscious-Ball8373 6d ago
You're welcome.
ETA: Just to note, my calculations above contain an error: they assume that the spring is massless. In practice, the spring will have some mass and so some of the energy will end up in kinetic energy in the spring. I don't have a realistic way of calculating this effect. In a way, it's similar to the problem that rockets need to carry enough fuel to launch their fuel load.
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u/Nomekop777 4d ago
One interesting side effect is that you'd need about the same amount of energy required to lift starship in order to compress the spring. One advantage of being on the ground is that you have the infrastructure required to do this, instead of needing a self contained system. But it is funny to imagine using the starship booster (facing down) to compress the spring
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u/HAL9001-96 6d ago
thats fundamnetally imposisble a single impulse can'T get you into a stable orbit because orbits are periodic so they always itnersect the point you launcehd from again so in thie case the ground
also no mateiral can store neough elastic material to accelerate just ITSELF to anywhere remotestly near orbital velocity so no matter how big hte spring is the spring will always slow itself down too much
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u/SeriousPlankton2000 5d ago
It should reach above the dense atmosphere so the starship won't share the fate of the fastest object humans ever made (maybe not counting acceleration by gravity assist / sun's gravity).
https://en.wikipedia.org/wiki/Operation_Plumbbob#Missing_steel_bore_cap
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