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Please consider this.

Chimborazo is a sleeping volcano at 01°28′09″ S 78°49′03″ W. Its peak is 6263 metres above the sea level and because it is very near of the equator, the peak is more than 2 kilometres farther from Earths gravity centre than peak of Mount Everest. And because it is so close to the equator, rocket launching platform there will gets almost all the benefits from the Earths rotation.

If we can build MAGLEV rails on the slope of the mountain and put a rocket on a carriage, and if speed of this carriage at top of Chimbarazo is 2000 metres in a second, what we can achieve with this construct?

2 km/s at 6200 metres above sealevel, 30 degrees angle upward.

So if we use this to launch Apollo 11, or some Shuttle mission or James Webb Space Telescope, how much smaller amount of rocket fuel we need?

That is the question.

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    $\begingroup$ There are a number of issues with the proposed catapult design, recommend an edit to move the question about fuel savings to the top and explicitly make the catapult launch 'magic'. removing mention of apollo and JWST will also avoid answers explaining why neither of them are likely cargos for such a system. $\endgroup$ Jan 29 at 11:42
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    $\begingroup$ Answers relevant to all the other issues of this concept other than the actual math being asked about space.stackexchange.com/questions/41135/… $\endgroup$ Jan 29 at 11:47
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    $\begingroup$ Fuel is cheap, supersonic trains that can carry multi-million pound loads are not. $\endgroup$ Jan 29 at 14:23

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In delta-v terms, your launcher is doing about 2/3 of the work of the Falcon 9 booster, so your vehicle will require around 1 km/s more performance than a Falcon 9 upper stage. Except with only about 4 km of slope available, it will need to achieve this while surviving ~50 gravities of launch acceleration and hypersonic flight in the dense atmosphere. Good luck with that.

The Falcon 9 upper stage is already relatively large, handling most of the delta-v of reaching orbit. Most likely, you will still need a two-stage vehicle. However, even a small booster stage will end up coming through a high energy reentry far downrange, and the upper stage will have a high mass overhead due to the added requirements of surviving the maglev launch, so both stages are likely to be expendable.

So, you're saving propellant, which is literally cheaper than dirt, at the cost of expending more high-performance aerospace hardware, which is not cheap. And you can only launch payloads capable of surviving the high-acceleration launch, and you can only launch them to an orbital inclination matching your maglev mass driver capable of slinging hundreds of tons up to 2 km/s. And you have to build and operate that 4 km long hypervelocity mass driver on rugged volcanic terrain on a remote mountain.

Fundamentally, you're optimizing the wrong thing. A Falcon 9 launch with booster and fairing reuse is priced around \$50M, some launches have gone even lower. The propellant costs are around \$200-300k...less than 1% of the overall cost. If you eliminated propellant costs entirely without changing anything else, that's all you could reduce launch costs by.

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