So think of going over a hill that would be one mile if a person was to travel straight through it to the other side.
If travelling up a 5% grade and then down a 5% grade the traveller would actually travel 1.015 miles. If done at 60 MPH it would take 61 seconds to travel the one mile, instead of the expected 60 seconds. The drivers speed in the x-direction would only be 59.77 MPH. While this doesn't seem like much, over the course of many miles it can cause the assumption of a flat earth to yield substantial errors.So I kept my mileage from Ann Arbor to Seattle and then went into Google Maps to compare my car's mileage with the flat Earth mileage. Now I'm driving a new car and I believe the odometer to be pretty accurate. I added a handful of miles to the Google Maps mileage to cover the times I got off the highway. I found that the Google Estimate was 2,346 to my odometers 2,389 miles.
43 miles seems like a lot of elevation. In one shot it would take me well into the stratosphere and back down. But when you think of it as going over over 22 one mile high mountains it doesn't seem so that unfathomable.
But I think it wasn't just elevation that caused my odometer to be higher than the Google estimate. I started to think that if Google thinks the earth is flat, then it also isn't accounting for the curvature of the Earth. I wondered how much affect this had on the difference in mileage. I determined the radius of the earth at 45 degrees north to be about 2,805.8 miles. Ann Arbor is at 83 degrees west and Seattle is at 122 degrees west. This means that I traveled about 10.8% of the way across the world. Here is a diagram:
Google says it is 1883 miles, but actually it is 1903 miles with the curvature. That is just a straight line due west at the 45th parallel which is why it is much less than the actual route which twists and turns and goes north quite a ways. So I believe that 20 miles of the difference between my odometer and the Google estimate can be accounted for by the curvature of the Earth. This would also mean that I could account for some error due to the curvature of the Earth from south to north, but since I traveled less than 400 miles north I'm going to ignore that. That means for every 144.45 miles I drove, I had to drive an extra one to make up for the curve of the earth.
The final breakdown would be:
2389 actual miles
- 10 miles spent off of route
- 20 miles of curvature
- 2336 miles of Google route
Which leaves me with 23 miles of elevation changes. That is about the same as driving up and down the height of Mount Everest twice.
4 comments:
This is the kind of problem that high school algebra teachers love to give (and would love to know that their former students still solve).
I regret reading that post ever so much
i think you need to account for much more inaccuracy in the cars odometer. on "internet", i saw this:
5 Foreign Models Tested for Odometer/Speedometer Inaccuracy in 2001:
Subaru Impreza WRX 3.2%
Ford Focus 2.1%
Chrysler PT Cruiser 1.07%
Toyota Condor 2400 1.54%
Volvo V70 T5 0.14%
that's pretty significant. add to that the fact that, with every turn you take, you rear tires are cutting the corner, and traveling a modestly shorter distance than your front tires. your speedometer/odometer is generally hooked up to your front tires, which gets you a different number than the rears.
Odometer error of 1% would account for 28 miles of difference between actual and Google mileage. This would make the difference only 15 miles. But if my odometer was off by 1% in the other direction it would make the difference 58 miles.
If I had some way to determine the change in elevation that I went over then I could back-solve for the odometer error.
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