Why do spaceships have a top speed?

[Kalessian]

I have a number of questions about physics that bother me. I mainly want to understand the limits of accelerating; why you can't just start throwing crap out the back of a vehicle and never stop going faster. I have a pretty good idea of each one, but I don't have the math to back up my thoughts.


First, I want to be able to determine the acceleration of a space ship in a vacuum given its specific impulse and its mass. We should assume the mass of the fuel is negligible, so no need to involve a lot of calculus

Now, let's take a modern chemical rocket. It has a specific impulse of 450. As I understand it, this means that 450kg of thrust is produced, for one second, for every 1kg of fuel consumed.

Since thrust = mass of the exhaust x accel. of exhaust, you get F(t)= M(e) x A(e)

Every action has equal reaction: magnitude of F(t) = Mass of Ship x Accel. of Ship?

My problem is with specific impulse and thrust, here. I want to measure thrust in Newtons, but specific impulse is giving me a mass...

But with thrust in N, I could get the acceleration of the ship if I also knew its mass, correct?

Now here is my big problem. I'm pretty sure there's a sqrt function in here, somewhere. I know the rocket's velocity can't just keep increasing - this isn't a constant acceleration problem. I imagine the rocket getting closer and closer to its max speed but never reaching it. Like at some point you hit really deminishing returns, and you can't throw stuff out the back of the rocket fast enough to propell the ship faster. I'm pretty sure it's closely related to exhaust velocity but I'd like this explained.

What's the mathematical model for this? How can I find out what time it would take to approx.(say within 5% of the asymptote) reach the rocket's max speed?

Or am I totally wrong?

What I'd like is someone saying: ok, a rocket with mass of 10 tons and a SI of 450 would take t amount of time to reach its max speed. And an explanation, please.


Ok, so now the relativity part. Let's take a flat spacetime that's completely void and empty except for person A at rest and person B in his spaceship. Person B has with him an accelerometer and a clock. Person A has a clock. Their clocks are in sync when the ship starts from rest at A's position.

The ship moves, magically, at a constant acceleration. No concerns about thrust or anything, the net force that moves the ship is external, magic, infinite. Now, as the ship approaches .9c, according to A his mass is now 2.3x as large as it was before. Then at .999c, it's 7x as large. Because of relativity, the relative mass increases.

Also, at .999c, their clocks are no longer in sync. A's clock will be much farther ahead of B's clock, correct?

So I understand what happens according the A's point of view, but what about B's?

According to B, at .999c, his ship is not any larger than it was when he started. Is everything else, such as A himself, getting much smaller to B?

According to B, at .999c, his clock is still moving normally, he doesn't feel like he's in slow motion at all, but if he looked at A, would A appear to be fast-forwarded super-fast?

According to B, at .999c, what would his accelerometer read? Would it still be the same as before? I think it would be, but then how can I explain that B still hasn't reached c? Time hasn't changed for him, Mass hasn't changed for him, and Acceleration hasn't changed for him, then how can velocity not equal c and then exceed it? B MUST be noticing something changing...

Is it that time is changing so fast outside his ship that he THINKS he's going much faster at .9999c than .999c, when in reality he hasn't???

It's late and I've confused myself, I'll give up now.

 

[andyandyandy]

Holy jebus im confused..

 

[gsellis]

Approaching c is easy as working Einstein's energy equation shows that as speed approaches c, it takes infinite energy to create that velocity ("mass" appears to be infinite IIRC).

Also, do not assume that space is a perfect vacuum. It is not and there is resistance.

As for your original question, it has been 26years since I took that physics class and it was not my strength.

 

[imported_inspire]

Originally posted by: gsellis
Also, do not assume that space is a perfect vacuum. It is not and there is resistance.

Right, and the force of drag is directly proportional to the square of velocity.

 

[TK2K]

Okay, i dont know mathematically, but i can explain it conceptually.

First, there is the question of thrust. Lets say you had a ship that could withstand very fast acceleration without coming apart. Even if this ship had a fuel tank the size of the moon it could never hit even half of light speed. the reason is quite simple, although the ship does not have weight, or very little weight, it does have MASS. and since mass is a constant property, all laws that effect it still apply.

therefor, to move a 12 mass object, you need to exert a force of at least 12.001 mass onto it. People think "oh i could lift 10,000 lbs in space" but you only could if you had something solid to push off of, otherwise your force would just push you away.

this is partially why ships have a top speed.



To get furtehr into it, lets say your object is accelerating at a given rate. It will slow down as gravitational forces act upon it, ect. Also, in order to throw stuff out the back, you would have to cary that stuff, thus the amount of stuff you woul dneed to throw out increses, ect ect ect see?

Now, for your relativity thing.


person in ship A would think he is travling at a normal time rate, and person in ship B would think the same thing about him. Person B would NOT see person A moving super fast, but would see something totally messed up. Think about it, lets say the conversion factor was 1 to 2. so 1 second in B is 2 seconds in A.

there now would be effectivly twice as much light from A getting to B then there would be if they were at relatively equal time. this could cause all sorts of funky things. From person a's point of view, ship B would get dimmer and dimmer.

this is one theory of dark matter, it is just normal matter traving so much "slower" that the amount of light it emits is spread out over such a long period of time we cannot really see it, but since gravity does not work in that manor, its all good.


i hope this helps

Originally posted by: andyandyandy
Holy jebus im confused..

come on man i live for this stuff

 

[Kalessian]

"therefor, to move a 12 mass object, you need to exert a force of at least 12.001 mass onto it. People think "oh i could lift 10,000 lbs in space" but you only could if you had something solid to push off of, otherwise your force would just push you away."


Ahah, that helps out a lot, conceptually.

As for relativity, I'm still not sure what to expect as person B, but I never thought about him getting dimmer to A because of the time/light emitted thing. That's cool stuff

 

[MetalStorm]

If you have a rocket in space and assume a complete vacuum and no gravitational effects then when the rocket uses its engines the acceleration will be very close to linear... However it's not the speed of light where the acceleration starts to tail off in conventional rockets, its the speed of the exhaust gasses moving away from the rocket. Let me give you an example:

Assuming gravity is not taken in to account and there is no friction:
If you have a rocket at rest which starts its engines you will get the maximum acceleration possible. As the rocket increases in speed, the acceleration will reduce not due to how close it gets to the speed of light but due to the DIFFERENCE in speed of it relative to the exhaust gasses being produced. Eventually when the rocket gets to the same speed as the exhaust gasses going the opposite direction, the acceleration will be 0. In fact you will find when you get to that speed the exhaust will be left as a trail of STATIONARY gas.

Hence the speed of rockets is down to how fast the exhaust gasses can be made to exit the diffuser.

 

[BrownTown]

Originally posted by: Kalessian
"therefor, to move a 12 mass object, you need to exert a force of at least 12.001 mass onto it. People think "oh i could lift 10,000 lbs in space" but you only could if you had something solid to push off of, otherwise your force would just push you away."


Ahah, that helps out a lot, conceptually.

As for relativity, I'm still not sure what to expect as person B, but I never thought about him getting dimmer to A because of the time/light emitted thing. That's cool stuff

How does an incorrect statement make sense conceptually?

Anyways, the reason that a ship in real life has a top speed in space is very simple. The ship can only carry so much fuel, and that fuel can only have some real energy density. So, the basic forumla of course F=MA, you can only apply a force for some small amount of time, so you can only accelerate during that time interval, and therefore after that time you are no longer accelerating and your velocity is constant. Alteratively you can think that your (mass of fuel) * (energy density of the fuel) only provide a certain amount of energy, since energy must be conserved the amount of energy released by your fuel is equal to the kenetic energy of the vessel after the fuel is burned, so energy of fuel = .5MV^2, and therefore the maximum velocity you can achieve is proportional to the square root of the energy provided by the fuel. This is basic high school physics, and is what matters in real life situations. The theoretically limit problem is obviously not high school physics, but its not that hard to get the basics either if you just don't ask why the equations work .

 

[RossGr]

According to my copy of Halliday and Resnick the velocity of a rocket is given by

V = - v ln(m/M)

Where:
V = final velocity
v = velocity of the exhaust gas relative to the rocket
M= Mass of Rocket without fuel
m = mass of fuled Rocket.

This shows that the maximum velocity is only limited by the ratio of fuel to payload. The exhaust velocity is not a limit because it is measured wrt to the rocket, it will always be large as far as the rocket is concerned.

 

[Kalessian]

I don't believe that's correct, though. If you had a reactor that could TELEPORT fuel from some infinite source onto the ship, then use it to propell the ship, you're saying that you would never stop gaining speed linearly?

So I can just use v=at to find its velocity at any time given a constant acceleration provided by my infinite fuel supply?

 

[Kalessian]

RossGr, thanks, that equation is very helpful.

Seems like I'm having trouble separating my own theoretical situation with what's actually practical.

Of course we're assuming speeds less than c to begin with.

I'm at least happy to see exhaust velocity in the actual equation


--Actually, I did some math out and things get interesting

Even in a practical situation, ie fuel is stored on the rocket like normal...

It IS closely related to exhaust velocity....

Watch... we said our chemical rocket has an exhaust velocity of 4.5km/s, right?

Now lets say the fuel to empty ratio is 4... which is pretty big, considering that means the ship is 4 times heavier with it's fuel than without, so it's a lot of fuel.

That's 1.38 times the exhaust velocity. So a max V of about 6km/s. Not much higher than that inital 4.5km/s, which is what you're going to keep running into.

Now that you mention it, I wasn't thinking of a sqrt function. It was an logarithmic one.

 

[MAW1082]

Originally posted by: Kalessian
"therefor, to move a 12 mass object, you need to exert a force of at least 12.001 mass onto it. People think "oh i could lift 10,000 lbs in space" but you only could if you had something solid to push off of, otherwise your force would just push you away."

Yeah that's not right at all. If you weighed 100 kg and you pushed on a 1000 kg weight, the speed you moved away from your initial point would be directly related to the speed the weight moved away in the exact opposite direction. Conservation of energy . . . for example

1/2*m1*v1^2 = 1/2*m2*v2^2

This is for two object initially at rest at a 'single' point. It's conservation of energy.

Think about that then think about reference frames and then think about space time then think about time dilation and all that junk.

 

[BassBomb]

correct me if im wrong but isnt space not exactly a vaccum, but its pretty close?

 

[Mday]

Originally posted by: BassBomb
correct me if im wrong but isnt space not exactly a vaccum, but its pretty close?

there are small particles and gases flying about.

 

[herm0016]

Originally posted by: inspire

Originally posted by: gsellis
Also, do not assume that space is a perfect vacuum. It is not and there is resistance.

Right, and the force of drag is directly proportional to the square of velocity.

actually the higher the speed the higher the exponent. so at normal speeds its a square, at interplanatary speeds it closer to being to the 3rd or 4th power.

 

[BrownTown]

To my knowldedge the drag on a spaceship in space is neglidgible for this calculation. Even with NO drag there is a maximum velocity that the spacesip can reach in REAL LIFE becasue you cannot just teleport more fueld on. Even in MAGIC LIFE there is still a limit, but that doesn not concern me becasue i don't beleive in magic .

Also, the equation which RossGr posted is just created by solving the conservation of mass. you just set the kinetic energy of the rocket equal to the energy of the exhaust and then do a little algebra and get the answer. I dunno, I have onyl taken high school physics and this all seems like a very strightforward question. Perhaps people here are trying to overcomplicate things like everyone always does in this "Highly Technical" forum. Whenever a physics question comes up the first thing you should always think is conservation of mass/energy/momentum/charge etc... This can be used to solve a HUGE number of problems. Most of the equation you see are just special cases of these different relationships.

 

[Vee]

Kalessian, BrownTown has the correct answers.

But it might be more interesting to consider not "fuel" but split it into "reaction mass" and "energy".

The only reason a space ship would have a max top speed is because it eventually runs out of reaction mass. As long as it has reaction mass it keeps accelerating.
(and all that about "drag" and "space is not absolute vacuum" is just fluff and need not be considered)

A high Specific Impulse means you need less reaction mass. Unfortunately it also means you need much much more energy. Which is why nuclear reaction propulsion is absolutely necessary for serious space travel.

 

[DrPizza]

Using SI units
Thrust is a force... measured in newtons, not kg.
The units of impulse are the same as the units of momentum: kg*m/s

BrownTown, the equation RossGr gave isn't simply solved as you said... you realize that ln means natural log, right?
His equation is the solution to a differential equation (calculus+ level mathematics); not that it's really difficult to solve. I understand why intuitively some people might think there's a maximum speed... "ahhh, so you want to go faster, huh? Well, by increasing the amount of fuel, you increase the mass of the rocket, and thus decrease the acceleration." And, this is true. However, since the mass of what's being accelerated is decreasing as fuel is burned, you have mass as a function of time. This will result in a more complicated equation that you've seen in high school physics. However, when solving this equation for when all the fuel is expended, you get RossGr's equation which clearly shows that there is no upper speed limit. (this is, obviously, ignoring relativistic effects which will be quite insignificant anyway until the velocity gets much closer to c) I'm assuming that RossGr's equation is correct; I'm not taking the time to work it out myself. (Now that I think about it, I believe this problem is in the calculus book I use in the section on solving linear differential equations; I'll have to check and see later on.)

Also, someone said that the energy of the fuel = the kinetic energy gained by the rocket. You forgot that the exhaust gasses have mass and are going to have kinetic energy as well. (And, throughout the flight, that velocity will be changing for the gasses exhausted at any point.) (Not to mention, the exhaust gasses will have thermal energy as will the spaceship)

 

[imported_inspire]

Originally posted by: MetalStorm
If you have a rocket in space and assume a complete vacuum and no gravitational effects then when the rocket uses its engines the acceleration will be very close to linear... However it's not the speed of light where the acceleration starts to tail off in conventional rockets, its the speed of the exhaust gasses moving away from the rocket. Let me give you an example:

Assuming gravity is not taken in to account and there is no friction:
If you have a rocket at rest which starts its engines you will get the maximum acceleration possible. As the rocket increases in speed, the acceleration will reduce not due to how close it gets to the speed of light but due to the DIFFERENCE in speed of it relative to the exhaust gasses being produced. Eventually when the rocket gets to the same speed as the exhaust gasses going the opposite direction, the acceleration will be 0. In fact you will find when you get to that speed the exhaust will be left as a trail of STATIONARY gas.

Hence the speed of rockets is down to how fast the exhaust gasses can be made to exit the diffuser.

I liked this answer. Thanks.

 

[BrownTown]

Bah, someone posted that it was square root and then edited their post. Thought you could get around the integral if you jsut wanted the final condition, but you cant becasue the velocity in question is changing. So, yeah you have to use that equation, but it still means that in the REAL WORLD there is a REAL limit to velocity because you have to have a real payload, and as you can see your taking the ratio of the masses, so you need a huge amount more fuel than payload to increase the speed considerably. In real life this would be prohibitively expensive. Actually, I remember hearing somewhere that a rocket fueled by black powder would have to be more massive than the earth in order to escape the earths atmosphere! Obviously, this is imporactical, and the same is true for much more powerfull fuels, there is still a very real limit to the final velocity that happens at only a small fraction of the speed of light if you are considering using any propulsion systems that human have now or in the forseeable future. Also, we did actually do the differential equation way when i took physics in high school, but i was trying to simplify it for this special case, but alas it just didn't want to be simple .

Also, the point about the energy stored in the fuel equaling the energy of the projectile is only valid when you also consider the energy in the exhaust gases. But this just means that your final velocity is even LESS than what would be predicted otherwise, so you still have the limited velocity in real world. The point still remains that if you have some finite energy source than you cannot expect to accelerate forever.

Originally posted by: inspire

Originally posted by: MetalStorm
If you have a rocket in space and assume a complete vacuum and no gravitational effects then when the rocket uses its engines the acceleration will be very close to linear... However it's not the speed of light where the acceleration starts to tail off in conventional rockets, its the speed of the exhaust gasses moving away from the rocket. Let me give you an example:

Assuming gravity is not taken in to account and there is no friction:
If you have a rocket at rest which starts its engines you will get the maximum acceleration possible. As the rocket increases in speed, the acceleration will reduce not due to how close it gets to the speed of light but due to the DIFFERENCE in speed of it relative to the exhaust gasses being produced. Eventually when the rocket gets to the same speed as the exhaust gasses going the opposite direction, the acceleration will be 0. In fact you will find when you get to that speed the exhaust will be left as a trail of STATIONARY gas.

Hence the speed of rockets is down to how fast the exhaust gasses can be made to exit the diffuser.

I liked this answer. Thanks.

Gah, you seem to like the wrong answers more than the right ones , listen to Dr.Pizza...

 

[MetalStorm]

BrownTown, maybe you don't understand what I said in my post so I shall try again.

The rocket is at rest.
The rocket starts its engines, lets just say for the sake of example the exit velocity of the exhaust is 10,000 m/s

So for example when the rocket is traveling at 1000m/s the speed of the exhaust will be 9000 m/s. Not relative to the rocket but to an outside observer.

When the rocket is going 6000m/s the exhaust will be traveling at 4000m/s. As the rocket reaches 10,000m/s the exhaust will be laid as a STATIONARY trail.

 

[JRich]

Originally posted by: BassBomb
correct me if im wrong but isnt space not exactly a vaccum, but its pretty close?

Let's call it very low pressure

 

[BrownTown]

Originally posted by: MetalStorm
BrownTown, maybe you don't understand what I said in my post so I shall try again.

The rocket is at rest.
The rocket starts its engines, lets just say for the sake of example the exit velocity of the exhaust is 10,000 m/s

So for example when the rocket is traveling at 1000m/s the speed of the exhaust will be 9000 m/s. Not relative to the rocket but to an outside observer.

When the rocket is going 6000m/s the exhaust will be traveling at 4000m/s. As the rocket reaches 10,000m/s the exhaust will be laid as a STATIONARY trail.

what matters is its speed relative to the rocket, not relative to some random guy in the middle of space. Just a little common sense should tell you that you are wrong.

 

[TuxDave]

Originally posted by: MetalStorm
BrownTown, maybe you don't understand what I said in my post so I shall try again.

The rocket is at rest.
The rocket starts its engines, lets just say for the sake of example the exit velocity of the exhaust is 10,000 m/s

So for example when the rocket is traveling at 1000m/s the speed of the exhaust will be 9000 m/s. Not relative to the rocket but to an outside observer.

When the rocket is going 6000m/s the exhaust will be traveling at 4000m/s. As the rocket reaches 10,000m/s the exhaust will be laid as a STATIONARY trail.

Yeah? And? So let's take your example. Before the exhaust is fired, it's sitting in a fuel tank moving 10,000 m/s to the left. After getting fired, it's now moving 0 m/s. To get the fuel from 10,000 m/s to 0 m/s requires a force to be exerted onto the fuel to get that negative acceleration. When the ship exerts a force onto the exhaust, the exhaust will exert an equal and opposite force back onto the ship. Since the ship was subject to force opposite the direction of the exhaust, it will go faster.

 

[DrPizza]

Originally posted by: BrownTown
Gah, you seem to like the wrong answers more than the right ones , listen to Dr.Pizza...

Not me... RossGr had the correct answer here
And, thankfully he did; all my physics books are on a shelf at school.