The hyperloop, Elon Musk‘s idea to make a high-speed transit system for people, has been an exercise in, well, hype. Despite a roster of companies and piles of VC money dedicated to realizing the idea—in which pods travel through a nearly airless tube at supersonic speeds—no human has taken a ride to date. But they’re not the only ones on the case: Musk is holding an ongoing competition for (mostly student) teams to build and test their designs. Last weekend, 700 members of 25 teams gathered in Hawthorne, California to see whose pod could clock the highest speed in the near-vacuum tube Musk built on the SpaceX campus.
The video above is from the WARR Hyperloop Team out of the Technical University of Munich. The students built a sleek little pod that clamps onto the metal rail running along the floor of the tunnel, runs on a rubber wheel, and uses a 75-horsepower electric motor for propulsion.
This video also can serve as a great physics problem. What was the acceleration? How far did it travel? What about the acceleration during the stop? Of course, there were probably sensors aboard this prototype hyperloop pod, but I like to see what I can figure out by just using the video. So I’ll just use what’s in this video—plus the title, which says the team’s prototype achieved a maximum speed of 324 km/hr.
The data I really want to start with is the position of the car at different times. I can easily get time from the video by assuming it plays out in real time, but what about the position? For this, I am going to use the reflections from the ceiling lights. You might think they wouldn’t be useful reference points, since they’re not fixed in space—their apperance depends on the viewing angle from the camera on the hyperloop test pod. However, as the reflection reaches the bottom of the camera view the pod should be at some particular location. If the lights are evenly spaced, then these distance values for the pod should also be evenly spaced. For now, I am going to say the distance between two light-spots is 7.62 meters (but I have a feeling it might actually be 24 feet). Where did I get that number? I’ll get to that later.
Here is my first plot. This is the position of the hyperloop pod as a function of time.
As the pod starts off, it accelerates for some short period of time (the upward curve at the beginning). After that, it mostly moves at a constant speed before finally accelerating and slowing down to a stop (the downward curve at the end). Notice that in this plot, the car only travels 885 meters even though the hyperloop test says it is a 1.2 kilometer track. I think this is OK because the video shows plenty of track left at the end of the run.
What about the acceleration of the pod at the beginning of the run? If it has a constant acceleration, I can fit a quadratic equation to the acceleration part of the data (I used the first eight seconds). Looking at the coefficients of the fitting function, the acceleration would be 7.23 m/s2 or 0.74 g’s. This seems plausible as the hyperloop contest states a maximum acceleration of 1 g.
However, what I really want to look at is a plot of the pod’s velocity. I can get velocity data from the position data by taking a numerical derivative. The basic idea is to look at the change in position and change in time between each measurement. Of course there are some little tricks in order to get smooth data—so maybe I will just go over the details in a future post.
But here is a plot of the pod’s velocity (in one dimension) as a function of time.
There are two things I want to point out in this plot. First, take a look a the maximum velocity. This pod gets to a peak velocity of 90 m/s. If you do a unit conversion on this value you would find this is equal to 324 km/h. Yup, that’s the exact velocity that was stated in the video. In fact, I used this velocity to determine the distance between light reflections, so it should be no surprise that I get this value for the speed. Second, look at the acceleration during the stop. Since I have a plot of velocity vs. time, the slope of the graph would be the acceleration. Fitting a linear function to this portion of the motion says that the stop was 24.9 m/s2 or 2.5 g’s. That’s much higher than the acceleration during the speed-up.
In the end, there are really three things that you could use to calibrate this data: the total distance, the maximum speed, or the initial acceleration. Perhaps you can go back and adjust the scale so that the car travels exactly 1.2 km—that might be fun. Oh, don’t worry. I know that the WARR team probably has the exact measurements for these things, it’s just fun to try and get them from only the video.