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Could someone provide a graph of speed vs mass of a typical spacecraft - say, Soyuz or Dragon, as it rises towards LEO? If not graph, then a single fixed point of mass at speed of 2 km/s will do.

Rationale: Vactrain:

VacTrain

A segment of this train might be used to propel vehicles to about 2 km/s. A branch in the track to separate "communal transport" from a NASA launch facility, a section with a fast-acting airlock, the vehicle exits the vacuum tube at a friendly angle upwards, engages rocket engines, crosses the atmosphere and enters the orbit. We gain 2 km/s at initial stage very cheaply, but we are limited by the dimensions and tolerances (like curve angles) of the communal part used primarily for plain human transport. This limits the amount of fuel we can fit in the launch vehicle, and as result puts a total cap on how much we can get to the orbit (if anything at all).

Additionally, would this mode of launch be viable at all?

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The Low Cost Access to Space Project data on OTASKI analyzes trade-offs between mass and rocket thrust to launch a rocket from a balloon or an aircraft with payloads of less than 10kg. The LCAS aims to provide low cost and innovative methods for low cost orbital launch system.

Diagram of delta-V vs Rocket Mass

Graph of delta-V vs mass

A Rocket Mass model(on which the graph is based) using the rocket equation is provided in the form of C Program takes various parameters such as payload mass, overall thrust, length to diameter ratio, etc. etc. as inputs and produces dry/wet mass as result. It is modeled on SSTO rocket with one engine.

Rocket Mass Model: Rough Sketch of the rocket enter image description here

Additionally, the Vactrain model is of-course viable. One point worth noting while using the Vactrain would be to separate the communal segment with the rocket segment by means of an air-lock once the rocket takes its course. Without an airlock, In case of launch failure or accident inside the rocket segment, A rupture or flaw might propagate to the whole tunnel and would be disastrous.

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