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Deriving the Rocket Equation from First Principles

The Tsiolkovsky Rocket Equation (hereafter referred to simply as “the rocket equation”) models the ultimate velocity of a rocket given its mass, the mass of fuel, and the velocity of the exhaust. While it may seem at first to be a straightforward relationship, the fundamental problem of rocketry is that a rocket must lift its own fuel. Say you want your 1000 kg rocket to accelerate to 1000 km/h, and you find that 200 kg of fuel has enough oomph to get 1000 kg up to that speed. So you fuel up your rocket, which now weighs 1200 kg, meaning it’s too heavy to get up to speed. So you have to add more fuel to lift the weight of the fuel. And again, and again and again…