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}