Maximum mass value possible to fuel a spaceship

[Daniel Sellers]

Homework Statement

We are given that a spaceship converts mass directly to light (emitted backwards) in order to accelerate. If an energy mc^2 has been converted to light ( from the Earth's reference frame) then what is the final speed of the ship (w.r.t. Earth) if the ship's rest mass is M and it's initial speed was 0 (again w.r.t. Earth.) Finally, what is the largest value of m (the rest mass converted to light) that's possible? This last question is what I'm struggling with.

Homework Equations

E = [gamma]mc^2
E = pc (for photons)
p = [gamma]Mv

(Perhaps another relevant equation I'm not realizing?)

The Attempt at a Solution

I have used both conservation of momentum (pc = [gamma]Mc^2) and conservation of energy (mc^2 = [gamma]Mc^2 - Mc^2) to write two equivalent expressions which give the same numerical answer for the final speed of this ship, given any mass for the ship and any amount of energy converted to light. Neither expression allows for a speed greater than c.

Neither expression shows that their should be an upper limit for m (mass converted to light), yet the assignment and my professor assure me that they should. My professor has stated that the reason for a maximum is the doppler effect (which I more or less understand) but that the math showing an upper limit for m does not depend on the doppler effect at all, it's just an explanation for why it is so.

My expression for final speed of the ship is v = c*sqrt[1 - (M^2)/(M^2 + m^2)] and there is another expression which gives the same numerical answer but is no more illuminating to my question.

How do I show mathematically and/or explain physically that there should be an upper limit of mass used to fuel my awesome light-ship? I've tried setting v < c and solving for m only gives m > 0 which is not helpful because the expression takes care of the speed limit anyway. Would love even a starting place for how to find the upper limit for m (in terms of M, preferably).

Thanks!

 

[Orodruin]

You seem to be assuming that the rest mass is constant, but the problem seems to indicate that the spaceship is accelerating by converting part of its rest mass into light that is emitted as thrust.

 

[Daniel Sellers]

You seem to be assuming that the rest mass is constant, but the problem seems to indicate that the spaceship is accelerating by converting part of its rest mass into light that is emitted as thrust.

That wasn't how I interpreted the question at all, but I'll try that and see if I can come up with an answer.

 

[Orodruin]

Where else do you suggest that the mass comes from? Either it must be supplied externally, or the ship must use up its own rest energy. If it is supplied externally, you will also need to know what kind of momentum it carries with it when supplied. There really is only the interpretation that the mass converted is part of the ship as far as I can see.

 

[Daniel Sellers]

Where else do you suggest that the mass comes from? Either it must be supplied externally, or the ship must use up its own rest energy. If it is supplied externally, you will also need to know what kind of momentum it carries with it when supplied. There really is only the interpretation that the mass converted is part of the ship as far as I can see.

I suppose I hadn't considered where the mass was coming form since it's more of a theoretical question focused on the result of applying such energy in the form of thrust.

In any case, revising my equations so that they account for subtracting M - m suggests that the limit on m is simply M because otherwise the expression would involve the square root of a negative number; we'd be out of fuel. I'm fairly sure that's not the limit on m that this question is asking for.

 

[Orodruin]

Are you sure the rest mass after will be M-m? How does your energy conservation look?

 

[Daniel Sellers]

If we're converting the rest mass of the ship, then the final mass would have to be M - m, but I am revisiting my conservation equations to see if they still work.

At this point I need to talk with my professor and clarify if we are converting the ships mass or not. I will post here again when I know more!

 

[Orodruin]

If we're converting the rest mass of the ship, then the final mass would have to be M - m,

Why? Rest mass is not conserved in relativity.

 

[Daniel Sellers]

Perhaps that's the case, but in our class we haven't gotten to the effects of relativistic speeds on rest mass.

Like I said, I'm going to need to clarify this problem before I try to solve it anymore. I still don't see why there should be any limit on the mass converted that has anything to do with the dipper effect.

 

[Orodruin]

Perhaps that's the case, but in our class we haven't gotten to the effects of relativistic speeds on rest mass.

There are no effects of relativistic speeds on rest mass. Would you mind sharing what educational level this question is posed at?

I still don't see why there should be any limit on the mass converted that has anything to do with the dipper effect.

To be honest, it has more to do with relativistic kinematics.

 

[Daniel Sellers]

This is an undergraduate level class, and it's been explicitly stated that the reason for the maximum is the doppler effect.

The two expressions I mentioned earlier were using momentum and energy conservation, respectively. If the ship's mass M has to cotribute to the mass converted to light, then I'm not sure we're to begin except to replace M by M-m.

This however, does not give me two equivalert expressions for v. Or my algebra is bad, which is possible but I've been very careful.

So I'm not sure where to start again, just sure I need concrete clarification on this question. No answer I've come up with so far suggests a limit anyway

 

[Orodruin]

This is an undergraduate level class, and it's been explicitly stated that the reason for the maximum is the doppler effect.

Does that make it right? It is just a simple question of relativistic kinematics, nothing else.

If the ship's mass M has to cotribute to the mass converted to light, then I'm not sure we're to begin except to replace M by M-m.

Unfortunately, that would be wrong. The ship will have a rest mass after the conversion which is necessarily smaller than M-m. If you are assuming that it is M-m you will end up with inconsistencies (or m = 0 and v = 0). I suggest that you start from energy and momentum conservation. If you are using 4-vectors, this problem is really simple, if you have not introduced them yet, it will require slightly more algebra

 

[Daniel Sellers]

The doppler effect is not supposed to have anything to do with calculation, so whether it's the correct explanation is perhaps not so relevant.
We have just introduced the idea of 4 vectors, little to nothing about how to work with them.
I have been starting with momentum and energy conservation, using the expressions I stated in the OP.
Maybe I just need to get more comfortable with 4 vectors to solve this problem, because if neither M nor M-m are correct for the final rest mass than I have no idea what to use for the conservation equation.
I know how much energy is converted to light ; mc^2. But I don't know what the expression would be for the final kinetic energy if not [gamma](M-m)c^2 - (M-m)c^2. Likewise for momentum conservation.

I appreciate all you posts so far, they have revealed a lot of gaps in my understanding of this topic.

 

[Orodruin]

First of all, let us work in units where $c = 1$. It will simplify all expressions considerably.

I know how much energy is converted to light ; mc^2.

So you know how much energy is in the light ($m$) and you know the initial energy ($M$). How much energy is then left for the ship? There is no need for finding the kinetic energy at all in this problem.

Likewise for momentum conservation.

You know the momentum carried by the light and you know the total momentum (it must be equal to the initial momentum). So what is the momentum of the ship?

Once you have the energy and the momentum of the ship, how do these relate to the ship's velocity?

Edit: And finally, how do the ship's energy and momentum relate to its rest mass?

 

[Daniel Sellers]

OK before I go any farther please tell me if you see any major flaws here;

With c = 1, p = m (for photons)

So ship momentum = m
Initial energy for the ship is M, so after inverting m to light we have M - m.

One equation derived in my text gives v = p/E.

Is it as simple as saying v = m/(M-m) to find the velocity?

Thanks again for all your help so far, it is greatly appreciated

 

[Orodruin]

Is it as simple as saying v = m/(M-m) to find the velocity?

Yes.

 

[Daniel Sellers]

Ok, then that's more progress then I've made on this problem all weekend.
I can relate the rest mass as M(r)^2 = p^2 - E^2. Not sure how to use this to find an upper limit though on m though.

If I set m/(M-m) < 1 so that v is restricted to less than c, then I can say that m must be less than M/2. Is there more to it than that?

 

[Orodruin]

Is there more to it than that?

No. Also, check the signs in your expression for the rest mass and note that the rest mass approaches zero as $v \to c$.

 

[Daniel Sellers]

Ah yes I wrote that expression with the signs reversed.

Well, that's the last time I ever do relativity calculations without setting c to 1. That really did greatly simplify things.

I may show this thread to a couple of my classmates, I know this problem will generate confusion for others as well and this has been very helpful. Thanks again!