How Does Mass Conversion Impact the Dynamics of a Relativistic Spaceship?

[lotrgreengrapes7926]

A spaceship of mass $$m\ \text{kg}$$ is propelled by converting $$r\ \text{kg}$$ of its mass into energy every second. Assume no friction and a perfectly efficient system.

1) Find the velocity of the spaceship at time $$t$$.

2) Find its acceleration at time $$t$$.

3a) Find the distance it has traveled at time $$t$$.
3b) Find the distance it has traveled just before it becomes purely energy.

4a) Find how much time has elapsed with respect to the spaceship at time $$t$$.
4b) Find how much time has elapsed with respect to the spaceship by the time it becomes purely energy.

This isn't homework, but I just wanted to check that my methods were correct.

 

[Mindscrape]

What methods did you use?

You obviously want to use conservation of energy and momentum. I don't know if I understand 3b) and 4b) because under your proposed system the mass will decrease exponentially, so it will take inifinite time and what time is just before infinity?

 

[lotrgreengrapes7926]

The amount of matter being converted to energy is not proportional to the mass, it is constant. For example, for m=1000 and r=1, it will take 1000 seconds.
1) $$v=\frac{\sqrt{2rmt-r^2t^2}}{m}c$$
2) $$a=\frac{rm-r^2t}{m\sqrt{2rmt-r^2t^2}}c$$
3a) $$\displaystyle\int^{t}_{0} \frac{\sqrt{2rmx-r^2x^2}}{m}c\, dx$$
I don’t know how to integrate that.
3b) ??
I’m not sure about number 4.
4a) $$t^2-\frac{r}{2m}t^3$$
4b) $$\frac{m^2}{6r^2}$$