Tyranny of Spaceflight
From Rise: The Vieneo Province
As explained by rana
We are JUST going to talk about acceleration and deceleration in the vacuum of space. When you add Atmospheric drag and gravity pulling against you are opening a whore can of new worms.
Lets say for easy you have a perfect fuel container that can hold infinite fuel and never get any heavier its self. Plus the perfect engine with no loss not even entropy. The usage of fuel is exponential. Lets explore it.
Load your magic rubber balloon full of fuel exactly enough to to get you to 0.5 the speed of light... Great now you can get to the closest star in just 7 years. But you would just plow on past at 0.5 the speed of light so you need to decelerate. But really the only thing you were accelerating was the weight of the fuel, in fact your rocket is just that magic balloon. Ok.
You need to accelerate the fuel for deceleration one unit of fuel to accelerate you would ONLY get you to 0.25 the speed of light. So you need 2 times the fuel to reach 0.5 Speed of light and 1 to decelerate. For a total of 3 units of fuel.
But you want to come home. So you need 3 units of fuel to accelerate the rocket 2 units to decelerate. 1 to re accelerate back home. But you now will hit earth at 0.5 the speed of light. So this flight needs 6 units of fuel. But you die when you hit earth.
We want to stop at earth not die... 6 units of fuel to accelerate the rocket away from earth. 3 units to decelerate at you destination 2 to re accelerate back home. 1 decelerate and not to die in a cosmic fire ball.
EVERY inefficiency, loose, extra bit of weight, adds more, and more, and more fuel you need. Every course correction or change in speed adds more and more fuel you need. Plus you can not stuff infinite fuel in the balloon. For a round trip you need to pack 12 times the fuel for each maneuver.