### Hyperloop Physics Questions and Answers: Calculating G-Forces

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.

By Rhett Allain
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