Modeling Rocket Thrust in the xkcd What If-World
I’m still working on my PhD in xkcdology. That means that I play around with this What If? world and try to figure out how it works. Oh, you don’t know about this world? There’s really not much to know except that it’s a browser game that gives you a rocket to move around and explore stuff. It’s fun.
Here’s what I have found out so far.
- Starting with just the first planet (I haven’t gotten to other ones), the gravitational field seems to be piecewise constant.
- When r is less that 45 meters (based on the size of person) g = -5.5 m/s².
- When r is between 45 and 51, g = -4.6 m/s².
- For greater values of r, g = -1.46 m/s².
- Oh, the length of the spacecraft if 1.702 meters (that’s useful for when you can’t see the person at the beginning).
OK, let’s look at some more stuff. What kind of thrust does the rocket produce?
Thrust and Acceleration
Here’s the plan. I’m going to get the rocket in “deep” space and mostly stationary (not trivial). Then, I’m going to fire the rockets and measure the acceleration. Of course I’m going to do a screen capture of the motion and then use Tracker Video Analysis to get the position of the rocket as a function of time.
The game keeps the spacecraft in the center of the frame, so here’s a frame corrected view of the motion (just for fun).
You can really tell that it’s zooming along this way. The rocket mostly moves in the x-direction, so here is a plot of the x-position as a function of time.
The part with the rockets on, it seems like it has a constant acceleration (since the position vs. time is parabolic). Fitting a quadratic function, the “A” coefficient is half the value of the acceleration. That puts the rocket acceleration at 14.4 m/s². Nice.
Oh, wait. There is a slight acceleration in the y-direction. Here’s a plot of the y-motion.
From this, there is an acceleration of 0.78 m/s². This gives an acceleration magnitude of 14.42 m/s² (pretty much just in the x-direction).
Maximum Velocity?
Can you just keep increasing in speed forever? What about for just a long time? Let’s find out. Notice that this gets complicated since the rocket doesn’t always go in a straight line (after interacting with other objects).
It seems like I found some “empty space” to get the following plot (something happened to Tracker, so I exported it to plotly).
This looks very linear with a slope of 78.6 m/s and a y-intercept of 6.997 m (which doesn’t really mean much). But that seems to be the answer. There is a maximum speed in this universe.
What About Net Force?
The rocket starts off on the surface of a planet. So, if this were a real world with real physics then there would be two forces on the rocket while the engines are on. There’s the downward gravitational force and the upward pushing thrust force. Yes, I don’t actually know either of these forces — but I do know the gravitational field (g) and the thrust force per mass (let’s call that f_r — it’s lower case since it’s not a real force).
OK, why did I call the thrust force F_T and the force per mass f_r? Who knows? I regret my naming decision, but I’m not going back. I’m just going to live with it. But the point is that there should be a relationship between the acceleration during take off (which I can measure), the gravitational field (did that) and the rocket thrust (per unit mass).
Here is the launch.
And here is the position-time plot.
It looks quadratic. That’s nice. From the fitting data, it has an upward acceleration of 23.4 m/s². If the gravitational field is g = 5.5 m/s², then this means the thrust per mass would be 28.9 m/s². OK, that’s odd. It’s twice the acceleration value in deep space.
I guess I’ll just have to add that as a special force rule. When the rocket is near the surface of a planet, it has twice the thrust. Fine. It’s not real physics, so I guess I have to accept that — at least for now.
Update
Here are my other posts about the What If World.
