- #1
Daniel Sellers
- 117
- 17
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!