Why does the speed of the propellant limit the speed of a space ship in open space?

Isn't speed a relative thing in space? If so, why would the speed of a propellant matter? Why can't a space ship accelerate infinitely?

-
You're asking about the calculation for a finite amount of fuel, right? – David Z Jun 30 '14 at 2:33
Imagine that the spaceship is 1/2 fuel, 1/2 useful cargo. Those pieces ultimately get divided. Then by momentum conservation, the final speed will be exactly like the average propellant speed, but in the opposite direction. If the fuel is a higher fraction of the mass, one may achieve a higher speed of the useful cargo than the propellant, but this increase is only logarithmic. – Luboš Motl Jun 30 '14 at 4:48
This is why laser-pushed sails are the best option for an interstellar craft - You keep all the fuel at home, and you final velocity is limited only by the width of the Gaussian beam wavefront, the wavelength of that Gaussian laser beam, and the width, as well as the optical and thermal characteristics of the sail that you can afford. Also, compared with rockets, where your payload efficiency is around 1%, a laser-sail can give you payload efficiencies as high as 30% – lurscher Jun 30 '14 at 5:10
@this: The same can be said of a rocket employing baking soda and vinegar, if the fuel tanks are big enough. – Beta Jun 30 '14 at 9:10
This site can be a lot of help when understanding rocket science - projectrho.com/public_html/rocket/engines.php :) – Luaan Jun 30 '14 at 16:30

Why can't a space ship accelerate infinitely?

Because a space ship needs to carry fuel, and because that fuel needs to be contained in a fuel tank.

That need to carry the fuel needed to make the spacecraft accelerate leads to the very nasty ideal rocket equation,

$$\Delta v = v_e \ln \left( \frac {m_{\text{initial}}} {m_{\text{final}}} \right)$$

The initial mass is the final mass plus the mass of the fuel and oxidizer that will eventually be expelled. The final mass includes the mass of the fuel tanks, the spacecraft structure, and whatever tiny payload that can be carried on top of that. Another way to look at that logarithmic growth in change in velocity is an exponential growth in fuel mass.

More fuel means larger fuel tanks and more structure to hold everything together. A delta V of 2.3 to 3 times the exhaust velocity is doable. That corresponds to a vehicle whose mass is initially 90% to 95% fuel. Beyond that, there's not much hope for a single stage rocket.

What about a multistage rocket? While multistage rockets appear to escape the tyranny of the rocket equation to some extent, they don't. They just change where the tradeoff occurs, and they add their own brand of exponential growth problems. The Saturn V + Apollo spacecraft that brought humans to the surface of the Moon and back was essentially a six stage rocket. The vehicle that took off from the surface of the Earth was arguably that largest, most powerful machine ever built. The vehicle that splashed down in the ocean when the mission was over was a tiny little thing that could barely hold three humans.

The only way to escape the tyranny of the rocket equation is the age-old answer to the age-old question "Doctor, it hurts when I do this: «bonk»" The answer: "Don't do that then."

Unfortunately, we don't know how to avoid the nastiness of the rocket equation. The only viable option currently that doesn't require carrying fuel is a solar sail. However, you can't accelerate forever with a solar sail because sunlight drops as the inverse square from distance to the Sun. Beyond solar sails? Now you're in the realm of science fiction.

The above assumes Newtonian mechanics. The relativistic rocket equation is far, far worse than the ideal (Newtonian) rocket equation. If the ideal rocket equation is evil, the relativistic rocket equation is evil incarnate. The relativistic rocket equation is one of the answers to the Fermi paradox.

-
Surely nuclear power or antimatter isn't science fiction. – this Jun 30 '14 at 7:43
@this - nuclear power and antimatter don't change anything, an engine powered by them still needs to expel propellant in order to move forward, and the same rocket equation applies. The propellant needs to be stored in the rocket, it's finite, and the propellant speed (determined by the amount of energy that the nuclear or antimatter power source can provide) limits the spaceship speed. Currently used ion thrusters are examples of such engines, and they can be powered by nuclear reactors or antimatter once it's practical to do so. – Peteris Jun 30 '14 at 8:44
@this If you created photons, I assure you you had a source of fuel that lost mass in the process. Also, for all intents and purposes, antimatter-powered anything is science fiction. – Chris White Jun 30 '14 at 10:36
@this - The problem still exists. You have to carry (and consume) fuel to create those photons. In fact, photons are pretty much the very worst choice when it comes to rocket exhaust. The ratio of momentum squared to unit energy consumed to create that momentum is tiny with photons. – David Hammen Jun 30 '14 at 11:01
Fusion is not science fiction. Look it up. It powers our planet. A rocket consisting of mostly fuel mass also complies with our physical laws. Your definition of science fiction must be very ( very ) broad. – this Jun 30 '14 at 18:11

The maximum theoretical speed that a spaceship can reach isn't limited by anything (except the speed of light of course). However for a practical spaceship with a finite amount of fuel, the speed of the exhaust will set a practical maximum on the speed of the spaceship. This is because in order to accelerate to a higher speed, the spaceship would have to carry more fuel to begin with, but this additional fuel would increase the mass of the spaceship, making it even harder to accelerate. This relationship is exponential, which means for a reasonable rocket (one that you could actually build), the exhaust speed of the propellant sets a practical maximum on the final speed of the rocket.

If I recall correctly this practical limit is roughly twice the exhaust speed of the propellent. After this, the diminishing returns get too ridiculous.

-
Roughly three times the exhaust speed is perhaps a better number rather than twice, but it's the same concept. Three times exhaust speed means a vehicle that is initially 95% fuel. That's a tough but doable engineering problem. Twice exhaust speed means a vehicle that is only 86.5% fuel initially. That's a bit low. Modern rockets fall in the 90 to 95% range. That was being nitpicky. +1. – David Hammen Jun 30 '14 at 11:28

As David said, in a rocket the final velocity is given by a function that grows logarithmically (i.e: really slowly) with the ratio between the mass of the rocket with and without fuel (the so-called dry mass). So if you somehow make your fuel carrying capacity a hundred times larger than the mass of the empty ship, you only increased your final velocity by something which is of the order of 2. If you increase that ratio to a whopping 1000 times, you only increase your final velocity in a factor which is of the order of 3. As those ratio get that large, your ship begins to have several structural and material problems as basically you are trying to move a whale worth of fuel on top of a paper craft. A realistic limit for chemical rockets is about a factor of 6-8, and probably as much as 15 could be achievable with fission fragment rockets.

This is why laser-pushed sails are the best option for an interstellar craft - You keep all the fuel at home, and your final velocity is limited only by the width of the Gaussian beam wavefront, the wavelength of that Gaussian laser beam, the width, as well as the optical and thermal characteristics of the sail that you can afford. Also, compared with rockets, where your payload efficiency is around 1%, a laser-sail can give you payload efficiencies as high as 30%

-
-1, for two reasons. The first and lesser reason is that your numbers are off. Your 100:1 spacecraft achieves 4.6 times exhaust velocity, and for your 1000:1 spacecraft the factor is 6.9. Do the math. The larger reason is your last paragraph (continued). – David Hammen Jun 30 '14 at 11:44
As far as "laser-pushed sails" are concerned, that's pure science fiction for now. Beam propulsion is but one of many perpetually low TRL (technology readiness level) technologies. It's impossible to say which, if any, of those perpetually low TRL technologies is the "best". One of those technologies might eventually break through the low TRL barrier, but sans a magical crystal ball, there's no foretelling which one will do so. – David Hammen Jun 30 '14 at 11:46
4.6 / 6.9 = 2 / 3, Which is why I said "of the order of", instead of "exactly", the other factor is constant in both cases – lurscher Jun 30 '14 at 16:22
TRL is irrelevant as the OP is asking about how to achieve acceleration that is unbounded by the Tsiolkovsky equation. Regardless of what your feelings are toward laser-pushed sail readiness, we use what we know of physics as it currently stands as best as we can, and what current physics says is that laser-pushed sails are the most feasible solution for interstellar flight, like it or not – lurscher Jun 30 '14 at 16:26
Given the existing options for interstellar craft technologies, laser-pushed sails certainly stand as the most feasible. Since all interstellar craft designs are currently low TRL, TRL is irrelevant on this case – lurscher Jun 30 '14 at 16:57