I am working on a game idea that involves (special) relativistic mechanics instead of classical mechanics. Working out the details that I needed was easy enough if I assumed that:
- ships had a maximum speed at which they could move relative to my base frame
- ships could instantly go from stopped to maximum speed or vice-versa
I didn’t like those assumptions at all. So, I started playing with the equations for relativity. In classical mechanics, the rate-of-change of velocity equals the force you’re applying divided by your mass: 
In special relativity, your mass increases with velocity. So, that equation becomes: 
For the purposes of this post, I’m going to assume the simplest initial conditions: you start motionless and at the origin. For ease of notation, let 
I have solved this problem at least thirty times in the past two months. Sometimes I used the simple initial conditions as above. Sometimes I did it in all of its gory details (including the messy case where the applied force is not aligned with the current velocity).
I got the same answer (well, plus the extra mess when I did the full-on problem) every way that I tried it.
So, why did I do it over and over again?
If this were classical mechanics, the end equation would have been 
I knew what the graph 
My assumption was that since 
Boy was I wrong. Here is a plot of the two together (created with fooplot). The red line is the classical mechanics version. The black line is the relativistic version. Here, the force is such that the body is accelerating at a rate of the speed of light per second
so they’ve already gotten up to around 28,000 miles per second before you can see any separation in the graphs here.
Definitely, I can see the resemblance. Now, to fix my intuition about square-roots.
Interestingly, the
looks eerily close to an approximation that I once saw for two-dimensional distance. Doing a little algebra, one comes up with 
.
The 2-D distance approximation formula was that
. The relativity/classical approximation above seems to be 
for this early portion where 
.
I really like the idea of a game where the speed of light is 1 m/s or whatever. It’s basically a Gamow’s “Mr. Tompkins in Wonderland”, the game. It took me an embarrassing long time to find the name of this work considering I work in a building named in his honor. If you haven’t read it, it might give a good foundation for all of the concepts involved without reading something more in depth. There is a copy at http://www.arvindguptatoys.com/arvindgupta/tompkins.pdf. I think the tricky bit will be the time dilation effects and the finite information propagation time.
If your curious, the approximation $\sqrt{1+a^2t^2} \approx 1+\frac{a^2 t^2}{2} – \frac{(a^2 t^2)^2}{8} + \cdots$ is a Taylor expansion for small $a^2 t^2$. I’m sure about that min/max stuff or where it came from. Remember that at a large enough value of $a^2 t^2$, the 1 under the square root becomes negligible and this should basically equal $at$ (a straight line like your intuition told you), not $a^2 t^2$.
I will check out “Mr. Tompkins”. Thanks.
And, yes, I knew that for big enough t, it had to be almost a straight line (with slope speed-of-light). I was just thinking that given what
looks like compared to 
would be relevant for 
compared to 
. And, so I thought that I was ending up with an equation where the instantaneous velocity at the start is unbounded and eventually comes down to the speed-of-light. I should have doubted my intuition long before the 30-th derivation.