How High Can You See the Curvature of the Earth
Traveling can be tedious sometimes. What happens when I get bored? I expect for interesting bug and calculations. Higher up you can see the drome terminal inside the Atlanta airdrome. If you happen to be in there during a low traffic time, it's quite impressive how long this corridor goes. I always wondered if you could apply this to measure the curvature of the Globe. Let's look at a few questions and estimations.
Is information technology straight or level?
In that location is a good gamble I am using these two terms incorrectly - but here's my definition. I am saying direct means that the floor is a linear function. If you lot shot a laser 1 mm above the floor at one end of the terminal, it would be 1 mm above the floor at the other terminate of the terminal. The other option is that the floor is level. For a level floor, the the ground surface would always be perpendicular to the Earth's gravitational field.
If the World were much smaller, you could easily run across the difference between these ii floor designs.
If I were building a super long hallway, I call up I would go far level instead of directly. Information technology just seems like that would exist easier to build.
How much does the Earth's surface curve over this distance?
Allow's suppose that the Atlanta terminal is level (past my definition). If I aim a laser such that it is right at the floor level and parallel to the ground at one terminate of the last, how much higher will it be at the other terminate of the terminal?
There are ii things to start with. Showtime, what is the radius of the World? This is actually a trick question. The Earth does non accept just i radius since it isn't spherical. Instead, the Earth is more like an oblate spheroid. It'due south wider at the equator than at the poles. Only for simplicity, permit's say that the Globe is perfectly spherical with a radius of vi.378 x 106 meters.
Adjacent, we need to know the length of one of the terminals. My picture shows concluding A, so let'due south utilise that one. If you apply the the classic version of Google Maps, there is a distance measurement tool. From that, I get a final length of 726 meters.
Now for some maths. If the World is a sphere, I tin depict a circle all the way around information technology. At present if I am continuing on the Earth and shoot a laser tangent to the surface, that would be direct line. I can represent both this circle and line equally equations (assuming the origin is at the center of the Earth).
If I solve for the y-value of the circle (in quadrant 1), I get:
The difference between y one and y 2 will give the vertical deviation betwixt a straight light amplification by stimulated emission of radiation and the curved Earth. But wait! This is actually cheating. This volition give the difference in the y direction, but perchance information technology should exist a radial difference. Of course, I am going to proceed anyhow - I suspect that for small distances the deviation betwixt radial and y distances volition be small. Also, there is only one horizontal variable in the two equations - x 2. I'll merely call this x. Here is the difference as a office of ten.
Only for simplicity, I chosen this deviation distance south. And so, what is the deviation value for a laser aimed across a "level" airport terminal? Putting in the value of 726 meters every bit well as the radius of the Globe, I get a deviation of 4.ane cm. Honestly, I am a niggling surprised. I thought the deviation would be much smaller than that.
Hither is a plot of the vertical deviation as a role of horizontal distance.
Remember, this is assuming that everything is perfect. Perfectly "level" floor and a perfectly spherical Globe.
How could y'all detect the curvature of the Globe?
Based on my adding higher up, it might actually be possible to measure the curvature of this concluding. My very first idea was to apply the tiptop epitome from within the terminal. If the terminal curves with the Earth, and then a line that forms the corner of the floor should also be curved.
You tin't see in this image, just I suspect that these dotted yellow lines would diverge from the line making the corners (if the hall is level). I doubtable it would be difficult to get a value for the radius of the Earth from this difference - simply at to the lowest degree you could see the Globe is curved.
The other option would exist the light amplification by stimulated emission of radiation arrow choice. Here's what I would do.
- Get 2 lasers and put them very close to the floor about 2 or 4 meters apart ane in front of the other.
- Aim the two lasers and then they both shoot down the terminal along the aforementioned line. Why two lasers? These two lasers together will assistance define the local tangent of the floor.
- Mensurate the height of the ii lasers above the flooring. This will be the reference value.
- Move down the concluding and measure the distance from the flooring to the laser. Subtract the reference value to get the deviation distance.
- Now plot the deviation distance vs. horizontal distance. It should be a function like the ane I plotted above. It'southward possible to use this information to observe the radius of the Earth. (I left off some steps in the graphing of the data - just you get the thought).
I retrieve that is a feasible experiment. I would just need the lasers and to get all the people to move out of the way.
Could you lot roll a bowling brawl all the way down the terminal?
If a light amplification by stimulated emission of radiation is too difficult to get past the airport security (but I remember they are allowed), you lot could perhaps bring in a bowling brawl. Actually, the whole bowling ball is important for another question that I haven't gotten to yet.
Could you roll a bowling ball so that it would make information technology all the way to the stop of the terminal? Really, I have no idea nigh the acceleration of a bowling ball on a floor like this. How about a quick experiment. Information technology just so happens that I have a bowling brawl and a hall.
I couldn't get a good side view of the brawl, so I just walked with information technology. You probably shouldn't watch this video, but here information technology is.
I can get the position of the bowling brawl past counting the squares that it passes over. Each tile is 12 inches long. Here's a plot of the ball's position.
Clearly, I demand more information to get a model of the ball's motion. However, I volition just go on with what I have. The dispatch of this ball is quite small, but if I fit a quadratic equation to the data I can get an dispatch of 0.0248 g/south2 (think that the dispatch is twice the t 2 coefficient). At present we just have a simple kinematics problem. How fast would I take to roll this brawl so that information technology travels 726 meters?
Time doesn't matter, so I will start with the following kinematic equation:
I already know the dispatch (well, it's the negative of the value above that I stated). The final velocity would be 0 one thousand/s (in the instance that it simply stops at the end of the terminal). I too know the change in x position - information technology'south 726 yard. Putting these values in, I go a starting bowling ball speed of half-dozen m/s (well-nigh thirteen mph). That doesn't seem too bad.
But how difficult would it be to aim the ball down the centre of the hallway so that it doesn't hit a wall? Conspicuously, if you lot basin perfectly downwards the middle with a perfect hallway, it will go all the way down. But what angular divergence in the initial velocity volition however make it to the finish? Imagine the hallway as a giant rectangle (because it is). Allow me summate the angular deviation such that the brawl starts in the eye of the hall and hits the end in the corner (so it just barely makes it downward). This diagram should assist.
This makes a correct triangle from which I can calculate this bending.
I just demand the width of the hallway. The map shows the width of the whole terminal, but there is stuff on the sides. I found this pdf map of the inside of Last A. Based on this, I have a hallway width of 9 meters. This would give an maximum angular deviation of 0.0062 radians.
Let's compare this to bowling in an actual bowling alley. An official bowling alley is 60 feet to the first pin (eighteen.3 k). The width of the pin is nearly 4.5 inches (0.114 m) at the widest bespeak. If you want to bowl a strike - maybe you have to hit that first pivot within a zone of iii.5 inches wide. Yes, I know bowling is more complicated than this, simply it's just an estimate. With this bowling aisle and target width, you lot would accept a maximum angular deviation of 0.0024 radians. Ok, that'southward helpful. It seems like it is more difficult to hit a bowling pin in the middle than to aim downwards a long airport terminal. I gauge it'southward possible.
Could you detect the Coriolis deflection of the brawl?
I originally started thinking about this long airdrome terminal while traveling. Of course I posted a movie on Twitter. Hither was an interesting response.
Yep, the terminal does appear to be aligned along the North-South direction. Why would the ball drift to the side? Well, I'chiliad not sure if you know this only the Earth is rotating. Since the Earth rotates, the surface of the Earth is an accelerating reference frame (we call this a non-inertial frame). Whenever you have an object in a not-inertial frame, you accept to add in false forces. For the case of an object moving closer to the axis of rotation in a rotating frame, nosotros telephone call this fake for the Coriolis force. Here is a basic description of the Coriolis strength and this is a much more than mathematical analysis of the Coriolis forcefulness.
In general, I can write the Coriolis strength as:
Here the Ω is a vector representing the angular velocity of the rotating reference frame (the Globe) and the v vector is the velocity of the object. Of course, the "x" is the cross product such that if the velocity is in the same direction as the angular velocity then there is no Coriolis force. Really, what matters is the component of the velocity in the direction of the axis. Atlanta is 33.7° above the equator, then if you lot are moving Northward then part of your velocity is towards the axis of the Earth (since the Earth is non apartment).
Ok, I am skipping the rest of the Coriolis details. If a bowling ball is moving North in Atlanta with a speed of 6 1000/south, it would have a sideways acceleration due to the Coriolis force of iv.48 x 10-4 m/s2. But is this pregnant? I think the best fashion to approach this question is to make a numerical model of the bowling ball equally it goes downwardly the terminal. However, allow me just guess. If the ball is moving half dozen yard/s and slowing down with a constant acceleration, I can summate the time of travel.
Using my estimated acceleration from the bowling brawl video along with an initial velocity of six m/s, I get a travel time of 241 seconds. Ok, now pretend that during this time the Coriolis acceleration is constant in both magnitude and management (which it isn't). I can calculate the horizontal deportation using the basic kinematic equation (since the initial position is zilch and the initial sideways velocity is cipher):
Putting my values in, I get a sideways motion of 13 meters. That seems meaning. But await! This is for a ball going 6 m/south the whole time (even though I used a changing speed to calculate the time). I gauge it could be significant if I did a more than realistic calculation. Really, I should just practice the numerical calculation of this.
Hither's what I would beloved to see. Showtime get a long East-West terminal and meet if we can roll a ball all the mode to the finish of the hallway. There shouldn't exist whatsoever Coriolis deflection in that instance. Then have the same ball in a Due north-Southward terminal and come across if at that place is a noticeable Coriolis deflection.
Maybe I should just carry around a bowling ball when I travel in case I encounter the perfect state of affairs to exam.
Homework: What would happen to this same problem on a smaller planet? How small would a planet accept to be to accept a very noticeable curvature in an airport terminal?
Source: https://www.wired.com/2014/03/see-curvature-earth-airport/
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