Is Elon Musk crazy or just awesome? This week, the serial CEO (Tesla, SpaceX, Neuralink) posted on Twitter about yet another one of his ventures, the super fast tube-based transportation system called hyperloop.
Is this even possible? Let's do some quick calculations. First, what is the speed of sound? I am assuming that Elon is referring to the speed of sound at sea level (and not speed of sound in a low pressure tube). In that case, a pretty standard value for the speed of sound would be about 340 m/s (760 mph). So half the speed of sound would be 170 m/s.
This means the hyperloop vehicle has to get up to a speed of 170 m/s and then back to 0 m/s in just 1.2 km, or about three quarters of a mile. If the acceleration for increasing speed has the same magnitude during the slow down phase, then it would reach its maximum speed right in the middle—after 0.6 km.
In order to calculate the required acceleration, let me start with the definition of acceleration in one dimension. It's basically a measure of how fast the velocity changes. I can write that as:
During the speed up phase, I know the change in velocity is from 0 to 170 m/s. However, I don't know the change in time. But don't worry, we can get this from the definition of the average velocity—which is a measure of the rate that the position changes (in one dimension). I know the change in position (0.6 km) and I also know the average velocity is going to be 85 m/s (0 m/s plus 170 m/s divided by two). That means the time to get up to speed has to be 7.06 seconds (and then another 7.06 seconds to stop). With this time, I can now calculate the acceleration to have a value of (170 m/s)/(7.06 s) = 24 m/s2.
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Is that a reasonable acceleration? Well, it's a little bit on the high side. Just consider this: If you took a bowling ball and dropped it off a building, it would accelerate downward with a value of 9.8 m/s2. This value is an important reference (also because of the way you feel on the surface of the Earth). We call this acceleration 1 "g". That means the hyperloop would accelerate at 2.4 g's.
If you accelerated in your car as fast as possible, you would be lucky to get an acceleration of 1 g. Or if you took your vintage Space Shuttle for a launch, you might get 3 g's or even higher, but not for very long.
So, if I had to guess, then this hyperloop test is just that—a test. Although a human can withstand 2.4 g's, there is no way a human could be expected to withstand that kind of acceleration while still playing on a smart phone or drinking a cocktail and eating peanuts.
Update: 7:30 pm Eastern 4/10/18 This story originally calculated the acceleration of a hyperloop pod to the speed of sound, not half the speed of sound.
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