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In space there is almost nothing to slow you down, and your proppelent/S will always be traveling relatively faster than your craft so you should accelerate infinitely close to the speed of light assuming you have enough proppelents. Why don't we though?

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    $\begingroup$ Your assumption is wrong. $\endgroup$ – Jasper Jun 2 at 20:21
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so you should accelerate infinitely close to the speed of light assuming you have enough proppelents.

That's correct, but only for unreasonably large values of "enough". In practice, it is not possible to carry enough fuel/propellant. It has to be stored in a tank, and building the structure for the tank is expensive and difficult.

The way to see this is via the Tsiolkovsky rocket equation $$\Delta V = v_e \ln \frac{m_0}{m_1}$$

Let's say that we want to reach 50% of the speed of light and we have a fuel that has an exhaust speed of $300\,000 \text{m/s}$ (faster than we have today, but not far from some technologies). Then we just rearrange to solve for the mass fuel fraction.

$$\ln \frac{m_0}{m_1} = \frac{\Delta V}{v_e} = 500$$ $$\frac{m_0}{m_1} = e^{500} = 10^{217}$$

This means to reach half the speed of light (ignoring relativistic effects) with a realistic engine, the ship has to carry $10^{217}$ grams of fuel for every gram of non-fuel (like the engines, the ship itself, etc.) That ratio is much greater than the ratio of the mass of the galaxy to a proton.

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  • $\begingroup$ Then you need a stage 2 somewhat smaller amount to slow down so you don’t smash into your destination at relativistic speeds. $\endgroup$ – 511mev Jun 3 at 2:09

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