Question about Accelerating to Nearly the Speed of Light

[Broomstick]

Although this centers around a fictional story idea of mine, the question involves hard science. Unfortunately, it also involves math beyond my current skills.

Hypothetically, our story involves as one part of the action taking a small spaceship, let's say it has a mass of 20,000 kg (making it about the size of the larger version of a Learjet, but maybe that's not relevant), and accelerating it to nearly the speed of light. Let's say about 95% of c.

- Assuming it took four months to accelerate it from a "zero" (let's say it's in orbit around a planet, any planet) to 95% of c at a constant acceleration what would be the g force imposed on the ship and its contents? I'm assuming "a lot", but that's a very vague measure.

- Assuming some sort of propulsion system that ran on a matter/anti-matter reaction (which, if I recall correctly, is theoretically how you get the most possible energy out of fuel) how much fuel would you need to accelerate that mass to those speeds?

- Since we're getting up to speeds where time dilation comes into play, what is the time differential between a "stationary" observer left behind on the planet and the spaceship at maximum speed? If one minute passes on board the spaceship how much time is that "back home" on the planet?

[Broomstick]

Oh, and I wouldn't mind an attempt to explain how to calculate these things so in the future maybe I can do this sort of thing on my own. I'm capable of learning, even if I suck at math.

[Jerry the Vampire]

Accelaration is delta-V/time.


Using F=ma the. Convert that in to terms of g (google it).



As for matter-antimatter. This isn't exactly the case as it gives off waste like everything else it just depends on how efficient the reaction is. It requires more energy to create that amount of antimatter, using current technology significantly smaller, like orders of magnitude.


If you want to work how the time dilation factor it's sqrt(1-(v^2/c^2)).

[Jerry the Vampire]

Oh and t=t(0)/(sqrt(1-v^2/c^2))

[Simon_Jester]

Hm. Jerry, don't we run into a bit of a problem calculating acceleration for relativistic rockets, because of increased relativistic mass?

Then again, I'm honestly too confused and distracted right now to sit down and rigorously work out whether that affects the acceleration experienced by the rocket- so since Broomstick is probably concerned mainly with the acceleration experienced by her crew, it may not matter.

---

If we just ignore the question of relativistic effects, we're looking at accelerating the rocket to 2.85*10^8 meters per second, in four months or roughly ten million seconds. So... about 28.5 meters per second squared, or about three gravities. Roughly.

[cosmicalstorm]

Poul Andersons story To Outlive Eternity has a similar theme. IIRC there is a point in the journey towards the middle of the story when their mass is so great that they suspect their gravity is destroying entire clusters of stars as they pass them by.

[Jerry the Vampire]

Hm. Jerry, don't we run into a bit of a problem calculating acceleration for relativistic rockets, because of increased relativistic mass?

Then again, I'm honestly too confused and distracted right now to sit down and rigorously work out whether that affects the acceleration experienced by the rocket- so since Broomstick is probably concerned mainly with the acceleration experienced by her crew, it may not matter.

---

If we just ignore the question of relativistic effects, we're looking at accelerating the rocket to 2.85*10^8 meters per second, in four months or roughly ten million seconds. So... about 28.5 meters per second squared, or about three gravities. Roughly.

Yes the change of apparent mass is dictated by m(0)/(sqrt(1-v^2/c^2)).


However I am a firm believer that if you are going to do Hard Science Fiction you should have more knowledge on the area than the post of a stranger on the Internet so I provided a simplified version so a "bench mark figure" could be derived. Of course I am tired so I could have misinterpreted the question.

[Jerry the Vampire]

Oh and I would have guessed 3G's as it would take something a year to reach c when under a constant one g of acceleration.

[Simon_Jester]

I'm not entirely sure it works that way- gravitational mass of a fast moving relativistic object would not affect things the same as just parking a very large mass in the middle of a place

The real issue is simply that it takes more thrust to go from 200,000,000 m/s to 200,000,001 m/s, than it does to go from zero to one. For a normal rocket we have a simple, well defined rocket equation. Fortunately, there IS a relativistic rocket equation now that I look for it... however, it is considerably more complicated.

This issue complicates the question of fuel expenditure and thrust required, if we want to calculate those.

[Jerry the Vampire]

Oh I'd be happy to do the calculation in the morning after a good amount of sleep I am just giving approximate ways so broomstick can use google to work it out.

[Jerry the Vampire]

Also note that the wiki article also talks about a matter-antimatter fuel source.

Broomstick, have you done any research? I can't imagine you wouldn't have found the wiki article.

[Kuroneko II]

- Assuming it took four months to accelerate it from a "zero" (let's say it's in orbit around a planet, any planet) to 95% of c at a constant acceleration what would be the g force imposed on the ship and its contents? I'm assuming "a lot", but that's a very vague measure.

If you have a constant g-force, then
[1] v/c = tanh(aτ/c),
where τ is proper time (measured on ship) and a is its proper acceleration, the g-force experiences by ship. So the answer is 5.3 g.

There is a very close analogy between normal (Euclidean) rotations and acceleration in STR. If, say, on a Cartesian (x,y)-plane, you take the point (x,y) = (1,0) and rotate it about the origin, you'll trace out a circle. If the rate of rotation is proportional to the arc length on that circle, then the slope from the origin to the (rotated) point is m = Δy/Δx = tan(αs), where s is the arc length and α is some constant of proportionality. In spacetime, it's just the same, except we get hyperbolas instead of circles, the 'slope' v = Δx/Δt is a velocity, and the length along a curve is the proper time of a clock going along it. A few comments on this analogy have been posted in a recent thread regarding the apparent paradox for two inertial observers with relative velocity, each one considers the other to be time-dilated.

The real issue is simply that it takes more thrust to go from 200,000,000 m/s to 200,000,001 m/s, than it does to go from zero to one. For a normal rocket we have a simple, well defined rocket equation. Fortunately, there IS a relativistic rocket equation now that I look for it... however, it is considerably more complicated.

There are many ways to think about this kind of problem, many of them confusing and a bit too easy to fumble. However, not all, and the end result is actually pretty simple and sensible. If we're talking about accelerating from rest of a rocket with payload mass fraction R = (final mass)/(initial mass) = m/m₀, then the Newtonian rocket has a final velocity of
[2] vNewton = -u ln R,
where u is the exhaust velocity of the propellant. The relativistic rocket has a final velocity of
[3] vSTR/c = tanh(vNewton/c).
In the limit of u→c, we get an ideal photon rocket, which is very easy to sanity-check through energy and momentum conservation in the stationary frame:
[4] vphoton rocket = (1-R²)/(1+R²).

- Assuming some sort of propulsion system that ran on a matter/anti-matter reaction (which, if I recall correctly, is theoretically how you get the most possible energy out of fuel) how much fuel would you need to accelerate that mass to those speeds?

You can get the required mass fraction from the above:
[5] R = exp(-atanh(v)/u)
where here the v,u are in fractions of lightspeed to simply the formula. The best you can do is a perfect photon rocket's u = 1, but any remotely drive, including M/AM one, is going to have an effective exhaust velocity far below that. For fusion, u ~ 0.10 is very optimistic. For an M/AM drive, I'm not sure, but there should be a lot of losses through neutrinos and various inefficiencies (e.g., rocket nozzle), so anything remotely close to 1 is probably very unrealistic.

However, note that the first question assumes constant acceleration, while the rocket equation and its accompanying fuel requirements assume constant thrust. Those are incompatible assumptions for a rocket.

[Simon_Jester]

I suggest that we begin by focusing on u (effective exhaust velocity, right?). If we can pick out some usable numbers for that, we're getting somewhere.

[Broomstick]

Jerry, I want to point something out. When you say:

Yes the change of apparent mass is dictated by m(0)/(sqrt(1-v^2/c^2)).

I'm sure that equation at the end actually means something to you, however, I never got that far in math. It simply wasn't required of anyone in high school way back in the paleolithic when I attended and indeed most high schools simply didn't offer a math that advanced if I'm understanding this at all (though I may not be). The vast majority of college students were never required to learn such things either. Since I didn't go into a STEM area I was able to obtain my four-year college degree without mastering such things. The bottom line is that it might as well be written in Swahili for all the good it does me.

This is why "just google it" doesn't work, I don't have the background to understand the succinct answer. In fact, I'm not even sure I understand the format because in my day we didn't type that sort of thing out, we used a pen and paper. So can we clear at least that up, please? After the "/" it says the "square root of (1 minus velocity to the power of 2 divided by c to the power of 2)" correct? I'm not sure I understand the "m(0)" part. Is "m" mass or something else? What do I do with the (0) part?

Yeah, I know, this is kindergarten stuff to a lot of you these days but I'm starting where I'm starting. Just to reassure you, I am familiar with “acceleration is delta-V/time” although I very much doubt I could calculate “delta-V”.

As for matter-antimatter. This isn't exactly the case as it gives off waste like everything else it just depends on how efficient the reaction is. It requires more energy to create that amount of antimatter, using current technology significantly smaller, like orders of magnitude

Yes, I'm aware there is no such thing as a free lunch and no matter-antimatter engine is going to be 100% efficient, the universe doesn't work like that. Regardless, matter-antimatter reactions have the potential to liberate the greatest energy per unit of fuel which probably what I should have said as engine efficiency is different than reaction efficiency or fuel efficiency or several other version of “efficiency”.

Oh and t=t(0)/(sqrt(1-v^2/c^2))

That is also largely opaque to me.

Hm. Jerry, don't we run into a bit of a problem calculating acceleration for relativistic rockets, because of increased relativistic mass?

You know, the increasing relativistic mass hadn't occurred to me, but that is one of the reasons you can't actually reach lightspeed, right? You're mass would go up to infinity and require infinite mass to accelerate further.

Good thing I'm not actually planning to get all the way up to c.

Then again, I'm honestly too confused and distracted right now to sit down and rigorously work out whether that affects the acceleration experienced by the rocket- so since Broomstick is probably concerned mainly with the acceleration experienced by her crew, it may not matter.

Yes, I am primarily concerned with the effect on the crew... although I'm planning the “crew” to be an AI. If nothing else, you wouldn't have to worry about life support like food, air, or water. Power would be needed, but if you're able to boost a rocket that fast I'm assuming supplying an AI with sufficient power to continue to exist would be trivial.

If we just ignore the question of relativistic effects, we're looking at accelerating the rocket to 2.85*10^8 meters per second, in four months or roughly ten million seconds. So... about 28.5 meters per second squared, or about three gravities. Roughly.

Hmm... Well, that sort of derails my plot point that an AI was needed due to the acceleration being far too high for a human being to endure even short term. Having experienced 3 g's myself, I'd have to say it's unpleasant but it's certainly tolerable short term. (Four continuous months might still be incompatible with a human, but I don't think sufficient research has been done on that). 3g's might be reliably lethal over time, but it's not the sort of dramatically lethal effect I was looking for.

Actually, it wouldn't be a bad thing if the time to get to the max speed is less for the crew – if I am understanding correctly the “about four months” is for the non-accelerating observer back on the home planet, correct? It would be subjectively less time for the “crew” on the rocket.

How long would it take to get up to 95% of c at, say, 20g of acceleration? (20g is usually considered fatal although some people have survived exposure to it.) How about 30g or 40g?

However I am a firm believer that if you are going to do Hard Science Fiction you should have more knowledge on the area than the post of a stranger on the Internet so I provided a simplified version so a "bench mark figure" could be derived. Of course I am tired so I could have misinterpreted the question.

I think what you misinterpreted was my level of mathematical education, which isn't at all unusual on this forum. To YOU that's a simplified figure. To me, it's not understandable. Partly, it's that you're writing in the equations here in a format that I'm not familiar with, as when I was making my stab at pre-Calculus we weren't allowed calculators in the classroom, yes, we were doing all of that stuff with pencil and paper still. I pre-date computers and calculators in the classroom and I still own an abacus and a slide rule (yes, really, I do own those things and even know how to use them though I do prefer calculators these days). Frankly, if calculators had been allowed in the classroom I probably would have done better, but hey, that's water under the bridge.

It also doesn't help that the last time I attempted any math beyond basic algebra was 35 years ago, I'm sure I'm even less capable of wading through the math than I was in high school due to lack of use.

Anyhow, I'm looking for sufficient understanding to know where to plug the numbers into the equations, along with sufficient understanding to plug the equation into a scientific calculator app (the default calculator on my computer, actually). This would allow me to make at least crude calculations for travel in space ships without looking like a complete idiot.

And yes, I should do more research than “the post of a stranger on the Internet” but there's nothing wrong with utilizing the help of a “stranger on the Internet” on the way to greater enlightenment.

The story doesn't require an indepth explanation with equations. It requires that I give time/acceleration/time dilation descriptions that someone capable of doing the math would say “oh, yeah, that's reasonable” rather than something you could actually use for navigating/traveling in space. No equations required in the story.

Here's a real world comparison – if I read in a story that someone drove down Interstate I-94 from Chicago to Detroit in six hours I'd say “huh, yeah, that's entirely reasonable” but if someone said it took them 12 hours I'd wonder what the hell the delay was and if they said two hours I'd go what the fuck, what kind of car goes 210 miles per hour not because I'd calculate their exact speed but because I know what the speed limit on the freeway is and how long it takes to get there, so someone doing it in 1/3 the time is clearly bullshitting me.

So, really, if you said “it would take about a year to accelerate to 95% of c” then you're guess of “oh, about 3g's to do it in four months” is actually closer to the answer I was looking for than your equation. Not that I have any problem with the equation (I did ask for it, after all) but I don't know how to work the equation.

Would it be reasonable to say (ignoring relativistic time effects for the crew and going by the home planet reference frame) that at 12g's it would take about a month to get to 95% of c? Except, of course, the ship's mass keeps increasing and increasing requiring more and more fuel and thrust to accelerate further...

[Kuroneko II]

I'm sure that equation at the end actually means something to you, however, I never got that far in math. It simply wasn't required of anyone in high school way back in the paleolithic when I attended and indeed most high schools simply didn't offer a math that advanced if I'm understanding this at all (though I may not be). ...

I recommend WolframAlpha, which is like an online calculator except much more powerful. I linked to it in the g-force figure calculated above, and what's there should make it really obvious how and what to plug in to vary the figures as you see fit. Enter something other than 4 months or the .95 for the fraction of lightspeed on that page to get whatever effect you want.

You know, the increasing relativistic mass hadn't occurred to me, but that is one of the reasons you can't actually reach lightspeed, right? You're mass would go up to infinity and require infinite mass to accelerate further.

Yes, although 'relativistic mass' is a mostly outdated term for something that's proportional to energy ( (relativistic mass) = (total energy)/c² ).

Hmm... Well, that sort of derails my plot point that an AI was needed due to the acceleration being far too high for a human being to endure even short term. Having experienced 3 g's myself, I'd have to say it's unpleasant but it's certainly tolerable short term.

Then good news: 3 g's is much too low. The actual figure is higher.

Actually, it wouldn't be a bad thing if the time to get to the max speed is less for the crew – if I am understanding correctly the “about four months” is for the non-accelerating observer back on the home planet, correct? It would be subjectively less time for the “crew” on the rocket.

If the 4 months refers to ship time, then it's 5.33 g. If the 4 months refers to the stationary frame, then it's 8.85 g.
That's considerably more uncomfortable even for a short duration and is at least very plausible that it's not survivable for humans over the time-scales involved here.

How long would it take to get up to 95% of c at, say, 20g of acceleration? (20g is usually considered fatal although some people have survived exposure to it.) How about 30g or 40g?

For rocket's proper time τ,
[1] v/c = tanh(aτ/c) ⇨ atanh(v/c) = aτ/c ⇨ τ = (c/a) atanh(v/c)
That's about 1 month.
[2] t = (c/a) sinh(atanh(v/c))
That's about 54 days.
Again, click the links and it should be obvious how to change any of the figures.

[Broomstick]

Thanks for the WolframAlpha link, I find that very interesting and potentially very useful. Being able to plug in the numbers into an equation really does help me. I did find manipulating the equations linked to be pretty intuitive.

Actually, it wouldn't be a bad thing if the time to get to the max speed is less for the crew – if I am understanding correctly the “about four months” is for the non-accelerating observer back on the home planet, correct? It would be subjectively less time for the “crew” on the rocket.

If the 4 months refers to ship time, then it's 5.33 g. If the 4 months refers to the stationary frame, then it's 8.85 g.
That's considerably more uncomfortable even for a short duration and is at least very plausible that it's not survivable for humans over the time-scales involved here.

Yep. Most (healthy) people should still be conscious at 5g's, particularly if reclined, but most also find it painful. My friends who fly aerobatics say that you get used it in a sense, you can build up some tolerance, but it always remains uncomfortable to painful. It puts a significant strain on a human body even if said body is lying down. A human under 5g's is either bedridden or confined to some sort of mobility device (wheelchair, power armor, whatever). God help you if you trip and fall while attempting to walk.

8g's+ isn't going to work for people. Most people are unconscious by that point. You'd have to live 24/7 in a g suit and even then you're pretty seriously impaired even if you can retain consciousness. I don't see that being practical. Which, for my story idea, is good news.

[Vehrec]

Hey Boomstick, I've found a mass-ratio equation, and a table of Time-dilation factors- the ratio is that for every day aboard ship, 3.2 days pass from a 'stable' reference that isn't moving at high speed relative to the rocket's point of origin. The mass ratio is a bit more complicated-it's basically e to a complex power that is composed of acceleration and time. So given 8 months of acceleration, because we want to stop, and assuming that 4 months is the ship time and Kuro's right about the needed acceleration being at minimum 5.33 g...

Well, with a 100% efficient engine that converts all it's fuel into photons for thrust, we're looking at a mass-fraction of fuel that approaches 39:1 And that 1 is everything-the payload, the crew, the engines. Everything. So if you have a 20,000 kg 'Learjet', then either all but about 2.5% would be fuel, or it would have a drop-tank of approximately 777,000 kilograms.

[Broomstick]

Well, hell, let's round that up to 800,000 kg of reaction mass because no conversion/engine/whatever is ever 100% efficient in this universe. Strap on boosters, right? Of course, the mass in one sense will steadily diminish as the fuel is burned off but the increased speed will increase the mass...

Well, for story purposes, let's say 3-4 months to accelerate at 5.3g (more or less) using 800,000 to 1,000,000 kg of fuel, most of it strapped on as boosters.

OK, here's the flip side - eventually our intrepid AI will want to return to the home base planet. And not as a meteor. That's 3-4 months at 5.3g (more or less) using 800,000 to 100,000 kg of fuel to de-accelerate. Hoo boy. And, of course, you'll need a fuckton MORE fuel to accelerate the return home fuel up to those speeds....

::: scratches head in thought :::

What if we don't have to take all the fuel with us? Bussard ramjet? Use an on-board engine to accelerate up to speeds at which a ramjet is moving fast enough to collect a reasonable amount of interplanetary or interstellar stuff, then use the ramjet to continue to accelerate. Reverse the process to de-accelerate for the return journey. As a bonus, the ramjet scoop might be able to be used as a solar sail close in to the star the planet orbits, and to help slow down on the return, a sort of solar parachute.

It would still require a fuckton of fuel, but it looks more practical.

[Imperial528]

Yeah, I know, this is kindergarten stuff to a lot of you these days but I'm starting where I'm starting. Just to reassure you, I am familiar with “acceleration is delta-V/time” although I very much doubt I could calculate “delta-V”.

Just a nit-pick, but I believe you have a bit of a mix-up in terminology. There are two delta Vs, really. There's regular change in velocity, which is a difference, and then you have delta-V in the terms of a rocket, which is basically the ability of the rocket to change its velocity from v0 to v1, and this is given by the rocket equation.

For the concepts involved in this sort of stuff (and a few timey-wimey things) I've recently been reading this book by Allen Everett and Thomas Roman. It goes into more depth more eloquently than most can express in a forum post, and there is an appendix at the back which contains all the math going on if your brain is ever in a masochistic mood.

WRT g-forces I've read (or heard) that you can survive many dozens of gs lying down.

Also, Jerry was referring to Lorentz Factor when he said this: m(0)/(sqrt(1-v^2/c^2)), though referring to it in the context of mass.

There are many ways of calculating and using Lorentz factor (λ) and I don't understand the more complicated ones, so I use the simple λ=1/√(1-(v^2/C^2) which is then assimilated* into standard Newtonian equations (Eg, F=λMA, KE=0.5λMV^2, and so on). For time dilation you can solve like so: Where local time is t1 and origin time is t0 then t1=t0*(1/λ).

Of course since I have yet to get a full handle on calculus my methods intentionally avoid it so they don't work well in changing reference frames, only in static end-results. Actually, since I brought this up, my excel workbook returns at an acceleration of ~5.33g a local time of 3.99 months and an origin time of 12.80 months. To the others: Is this correct?

*Because factored, integrated, etc. and other words I could think of are specific mathematical terms as well and I don't want to cause a confused ruckus.

Well, hell, let's round that up to 800,000 kg of reaction mass because no conversion/engine/whatever is ever 100% efficient in this universe. Strap on boosters, right? Of course, the mass in one sense will steadily diminish as the fuel is burned off but the increased speed will increase the mass...

Well, for story purposes, let's say 3-4 months to accelerate at 5.3g (more or less) using 800,000 to 1,000,000 kg of fuel, most of it strapped on as boosters.

OK, here's the flip side - eventually our intrepid AI will want to return to the home base planet. And not as a meteor. That's 3-4 months at 5.3g (more or less) using 800,000 to 100,000 kg of fuel to de-accelerate. Hoo boy. And, of course, you'll need a fuckton MORE fuel to accelerate the return home fuel up to those speeds....

::: scratches head in thought :::

What if we don't have to take all the fuel with us? Bussard ramjet? Use an on-board engine to accelerate up to speeds at which a ramjet is moving fast enough to collect a reasonable amount of interplanetary or interstellar stuff, then use the ramjet to continue to accelerate. Reverse the process to de-accelerate for the return journey. As a bonus, the ramjet scoop might be able to be used as a solar sail close in to the star the planet orbits, and to help slow down on the return, a sort of solar parachute.

It would still require a fuckton of fuel, but it looks more practical.

First I am obliged to admire your inner Kerbalness. Coincidentally, that image helps illustrate how fuel costs are so, well, huge.

Secondly, unless you plan on dropping your tanks, it seems like it may just be better to bring a refinery with you.

Thirdly, I have no idea if a magnetic scoop could work very well as a solar sail/parachute, since that would depend on the amount of charged particles being flung off, how much it oomph it can get out of them, etc. The main issue though is that you may have to go slower when going home, as the ramjet/sail/whathaveyou may not put out as much acceleration as your initial engines, and I don't think your AI would want to ram its ship into the sun at 10% of c.

[Vehrec]

Well, hell, let's round that up to 800,000 kg of reaction mass because no conversion/engine/whatever is ever 100% efficient in this universe. Strap on boosters, right? Of course, the mass in one sense will steadily diminish as the fuel is burned off but the increased speed will increase the mass...

Well, for story purposes, let's say 3-4 months to accelerate at 5.3g (more or less) using 800,000 to 1,000,000 kg of fuel, most of it strapped on as boosters.

OK, here's the flip side - eventually our intrepid AI will want to return to the home base planet. And not as a meteor. That's 3-4 months at 5.3g (more or less) using 800,000 to 100,000 kg of fuel to de-accelerate. Hoo boy. And, of course, you'll need a fuckton MORE fuel to accelerate the return home fuel up to those speeds....

::: scratches head in thought :::

What if we don't have to take all the fuel with us? Bussard ramjet? Use an on-board engine to accelerate up to speeds at which a ramjet is moving fast enough to collect a reasonable amount of interplanetary or interstellar stuff, then use the ramjet to continue to accelerate. Reverse the process to de-accelerate for the return journey. As a bonus, the ramjet scoop might be able to be used as a solar sail close in to the star the planet orbits, and to help slow down on the return, a sort of solar parachute.

It would still require a fuckton of fuel, but it looks more practical.

Oh man, that fuel fraction I quoted was for one round of acceleration and deceleration-you want two? I can compute two. Give me a second here...

Boomstick, you may want to sit down. Carrying ALL the fuel for accelerating and decelerating to make a round trip gives us a mass fraction of 1509.718889:1 That's uhhh....well, it's pretty much impossible to make a single-stage vehicle with mass fractions over 20. So you're going to need to asparagus stage a LOT of boosters.

Bussard Ramjets are unfortuneatly limited by their own drag on the interstellar medium they need for fuel to about .2c if I recall correctly. *checks* Ouch, worse than that, their top speed is about .12c! And they need to be boosted up to about half of that to get ignition in some parts of space.

[Vehrec]

Ghetto Edit: To be clear-the initial fuel fraction already assumed that the rocket would accelerate once and then bring itself to a stop at a destination, so the 1509:1 is for four rounds of acceleration one cycle to bring you out and one cycle coming back. And like any good rocket, most of the mass is in the first 'stage'.

[Broomstick]

Oh.... so if the idea is to accelerate, match velocity to pick up an object, then de-accelerate then you only need the original "mere" 800,000 kg of fuel, right? Because this isn't about going somewhere, stopping, and then going back, it's about intercepting a rapidly object in space.

[Darmalus]

Have you considered that the home base may be launching extra fuel for your ship to pick up mid-flight to decelerate/return home? If the ships just slows down a little bit, the deceleration boosters should be able to catch up and be caught at reasonable relative velocity. Everyone may be hurtling at 95% the speed of light, but the new boosters might be going at a sedate few meters be second compared to the retrieval ship.

[Jerry the Vampire]

Yeah, as long as you restrict space travel to governments/corporations you could give your spacecraft a "sail" and use a laser to push it on.

[Jerry the Vampire]

Also broomstick sorry about the maths I remembered learning it in secondary school (high school) but I have just remembered it was during 'A' Level Physics which is done when 16-18 and you would have had to taken it (not compulsory). Once I finish work I will work through it for you and post it.

Does anyone know if mike has scientific notation software on the board?