Editor's Note: The Hyperloop is a totally imaginary transportation device that has captivated entrepreneur Elon Musk, who keeps talking about it. First imagined at least 100 years ago, it would basically look like some version of those green tubes on Futurama.
The best part of the Hyperloop is that it's the coolest real-world physics problem. What's the Hyperloop? Who knows for sure, it is some type of transport that could get from LA to NY in just 45 minutes. Here is my favorite infographic on what we do and don't know about the Hyperloop.
The Hyperloop seems to be based on some other ideas about Evacuated Tube Transportation. Essentially, you get a big pipe and put some people in a pod-like device that goes in the tube. Pump most or some of the air out and then shoot the pod down the tube.
For the sake of this physics problem, let's make some assumptions (or guesses if you like).
- Reduced air pressure in the tube.
- Little or no friction on the rails due to magnetic levitation.
- A travel time of 45 minutes from LA to NY (a distance of 3.95 x 106 m).
- Maximum acceleration of 1 g (9.8 m/s2).
Now for the physics.
Practice With Graphs
Let me start with a graph. This shows the horizontal acceleration of the pod as a function of time.
Here the pod accelerates at 9.8 m/s2 for 2.6 minutes and then travels at a constant speed. At the end of its trip, the pod spends the last 2.6 minutes with an acceleration of -9.8 m/s2.
Question 1: Sketch a graph of velocity vs. time and another graph for position vs. time for this same trip. Be very careful. The common problem is to draw a velocity graph that LOOKS like the acceleration graph. However, remember the definition of acceleration and average velocity:
This says that the acceleration will be the slope of the velocity-time graph and the velocity will be the slope of the position-time graph. But in this case we are going backwards. It isn't too difficult to draw the velocity graph though. The graph should have a positive slope of 9.8 m/s2 for the first time interval, then it should have a zero slope for the next part. Of course, the velocity graph should be continuous - that would make the middle velocity constant (zero slope) and non-zero (so that it matches up with the previous interval).
What about the position graph? The first part of the velocity graph says that the slope of this position graph has to increase. This means that it would be a parabola. Or, if you like you can use the following kinematic equation.
Actually, that t should really be a Δt. But let me go ahead and show the two graphs for velocity and position that go along with that same acceleration graph above. Actually, let me change the problem. If the acceleration part is just around 5 minutes total out of 45 minutes, the curved part of the position graph is rather difficult to see. Instead, this pod accelerates for 7 minutes at the beginning and 7 minutes at the end.
For the position graph, many people would want to have the final position back at zero. Notice in this position graph, the SLOPE is zero at the end, not the position.
How Fast Would the Pod Go?
There are lots of ways to get from LA to NY. Let's look at a couple of different cases.
Case 1: 45 min Trip. There are reports that it would take 45 minutes for a the trip. If the acceleration is 1g, that would be 2.6 minutes for speeding up at the beginning and 2.6 minute of slowing down at the end. This would give an average speed of 1441 m/s (3223 mph) with a maximum speed of 1528 m/s (3418 mph). That is pretty darn fast. Just for comparison, the speed record for the SR-71 Blackbird is 2193 mph.
There is one problem with a case like this. The acceleration of 1 g seems fairly large for a passenger vehicle. Yes, you are at "1 g" right now but this would put additional stress on top of that for 2.6 minutes. What would this feel like? Let's look at the accelerating frame of the pod. Since the frame is accelerating, the actual acceleration would be a fake force in the opposite direction as the actual acceleration. That would make a force diagram like this:
The fake force and the gravitational force are the same magnitude, but perpendicular. This would give a net force of 1.4 g's at a 45° angle below the horizontal. So, you would feel like you are tilted back and about 40% heavier. Not too bad, but maybe not so great for all people. This doesn't see too much larger than the acceleration during a commercial airline take off - but certainly for much longer period of time.
What about the stopping part? In this case you would have the same g-force except it would feel like you are tilted forward. Humans don't take accelerations as well in this case - we do much better accelerating in the directions of our eyes. Again, this might not be too comfortable for 2 and half minutes. What if you have to use the restroom during this time?
Case 2: Accelerate the Whole Time. What if you accelerate half way there and then slow down for the second half? Here's what that would look like in terms of a velocity and position graph.
From this, you can see that there is a maximum velocity of 6223 m/s (13920 mph). Yes, this would be as fun of a trip since consist of a 10 minute acceleration (speeding up) followed by a 10 minutes of slowing down. There wouldn't be a drink service for that trip. I guess this could be the hurry up option. It sure would be cool to get to NY from LA in just around 20 minutes. Crazy cool.
Case 3: 1/2g Acceleration. Let's say you don't want to get too crazy with the acceleration. If you only want a .5 g acceleration, how long would the trip take? I'm not going to show the graph (since it would be boring). If the acceleration was still for 2.6 minutes, the trip would take 88 minutes with a max speed of 764 m/s (1709 mph). Still not too bad for a trip across the country.
This is a cool physics problem. I have only touched on the questions to answer. Expect more Hyperloop posts in the future.
