Traveling can be boring sometimes. What happens when I get bored? I look for interesting problems and calculations. Above you can see the airport terminal inside the Atlanta airport. 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 use this to measure the curvature of the Earth. Let's look at a few questions and estimations.

Is it straight or level? ————————

There is a good chance I am using these two terms incorrectly - but here's my definition. I am saying straight means that the floor is a linear function. If you 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 end 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 Earth were much smaller, you could easily see the difference between these two floor designs.

If I were building a super long hallway, I think I would make it level instead of straight. It just seems like that would be easier to build.

How much does the Earth's surface curve over this distance? ———————————————————–

Let's suppose that the Atlanta terminal is level (by my definition). If I aim a laser such that it is right at the floor level and parallel to the ground at one end of the terminal, how much higher will it be at the other end of the terminal?

There are two things to start with. First, what is the radius of the Earth? This is actually a trick question. The Earth does not have just one radius since it isn't spherical. Instead, the Earth is more like an oblate spheroid. It's wider at the equator than at the poles. Just for simplicity, let's say that the Earth is perfectly spherical with a radius of 6.378 x 106 meters.

Next, we need to know the length of one of the terminals. My picture shows terminal A, so let's use that one. If you use the the classic version of Google Maps, there is a distance measurement tool. From that, I get a terminal length of 726 meters.

Image: Google Maps

Now for some maths. If the Earth is a sphere, I can draw a circle all the way around it. Now if I am standing on the Earth and shoot a laser tangent to the surface, that would be straight line. I can represent both this circle and line as 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 1 and y 2 will give the vertical deviation between a straight laser and the curved Earth. But wait! This is actually cheating. This will give the deviation in the y direction, but maybe it should be a radial deviation. Of course, I am going to proceed anyway - I suspect that for small distances the difference between radial and y distances will be small. Also, there is only one horizontal variable in the two equations - x 2 . I'll just call this x. Here is the deviation as a function of x.

Just for simplicity, I called this deviation distance s. So, what is the deviation value for a laser aimed across a "level" airport terminal? Putting in the value of 726 meters as well as the radius of the Earth, I get a deviation of 4.1 cm. Honestly, I am a little surprised. I thought the deviation would be much smaller than that.

Here is a plot of the vertical deviation as a function of horizontal distance.

Remember, this is assuming that everything is perfect. Perfectly "level" floor and a perfectly spherical Earth.

How could you detect the curvature of the Earth? ————————————————

Based on my calculation above, it might actually be possible to measure the curvature of this terminal. My very first idea was to use the top image from inside the terminal. If the terminal curves with the Earth, then a line that forms the corner of the floor should also be curved.

You can't see in this image, but I suspect that these dotted yellow lines would diverge from the line making the corners (if the hall is level). I suspect it would be difficult to get a value for the radius of the Earth from this deviation - but at least you could see the Earth is curved.

The other option would be the laser pointer option. Here's what I would do.

Get two lasers and put them very close to the floor about 2 or 4 meters apart one in front of the other.

Aim the two lasers so they both shoot down the terminal along the same line. Why two lasers? These two lasers together will help define the local tangent of the floor.

Measure the height of the two lasers above the floor. This will be the reference value.

Move down the terminal and measure the distance from the floor 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 one I plotted above. It's possible to use this data to find the radius of the Earth. (I left off some steps in the graphing of the data - but you get the idea).

I think 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 roll a bowling ball all the way down the terminal? ————————————————————

If a laser is too difficult to get past the airport security (but I think they are allowed), you could perhaps bring in a bowling ball. 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 it all the way to the end of the terminal? Really, I have no idea about the acceleration of a bowling ball on a floor like this. How about a quick experiment. It just so happens that I have a bowling ball and a hall.

I couldn't get a good side view of the ball, so I just walked with it. You probably shouldn't watch this video, but here it is.

I can get the position of the bowling ball by counting the squares that it passes over. Each tile is 12 inches long. Here's a plot of the ball's position.

Clearly, I need more data to get a model of the ball's motion. However, I will just proceed with what I have. The acceleration of this ball is quite small, but if I fit a quadratic equation to the data I can get an acceleration of 0.0248 m/s2 (remember that the acceleration is twice the t2 coefficient). Now we just have a simple kinematics problem. How fast would I have to roll this ball so that it travels 726 meters?

Time doesn't matter, so I will start with the following kinematic equation:

I already know the acceleration (well, it's the negative of the value above that I stated). The final velocity would be 0 m/s (in the case that it just stops at the end of the terminal). I also know the change in x position - it's 726 m. Putting these values in, I get a starting bowling ball speed of 6 m/s (about 13 mph). That doesn't seem too bad.

But how difficult would it be to aim the ball down the center of the hallway so that it doesn't hit a wall? Clearly, if you bowl perfectly down the middle with a perfect hallway, it will go all the way down. But what angular deviation in the initial velocity will still make it to the end? Imagine the hallway as a giant rectangle (because it is). Let me calculate the angular deviation such that the ball starts in the center of the hall and hits the end in the corner (so it just barely makes it down). This diagram should help.

This makes a right triangle from which I can calculate this angle.

I just need 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 Terminal 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 (18.3 m). The width of the pin is about 4.5 inches (0.114 m) at the widest point. If you want to bowl a strike - maybe you have to hit that first pin within a zone of 3.5 inches wide. Yes, I know bowling is more complicated than this, but it's just an estimate. With this bowling alley and target width, you would have a maximum angular deviation of 0.0024 radians. Ok, that's helpful. It seems like it is more difficult to hit a bowling pin in the middle than to aim down a long airport terminal. I guess it's possible.

Could you detect the Coriolis deflection of the ball? —————————————————–

I originally started thinking about this long airport terminal while traveling. Of course I posted a picture on Twitter. Here was an interesting response.

@rjallain Do any of them align north/south? You could roll a ball down the middle and see if it drifts east/west. — Barry Fuller (@bfuller181) January 16, 2014

Yes, the terminal does appear to be aligned along the North-South direction. Why would the ball drift to the side? Well, I'm not sure if you know this but 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 non-inertial frame, you have to add in fake forces. For the case of an object moving closer to the axis of rotation in a rotating frame, we call this fake for the Coriolis force. Here is a basic description of the Coriolis force and this is a much more mathematical analysis of the Coriolis force.

In general, I can write the Coriolis force as:

Here the Ω is a vector representing the angular velocity of the rotating reference frame (the Earth) 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, so if you are moving North then part of your velocity is towards the axis of the Earth (since the Earth is not flat).

Ok, I am skipping the rest of the Coriolis details. If a bowling ball is moving North in Atlanta with a speed of 6 m/s, it would have a sideways acceleration due to the Coriolis force of 4.48 x 10-4 m/s2. But is this significant? I think the best way to approach this question is to make a numerical model of the bowling ball as it goes down the terminal. However, let me just guess. If the ball is moving 6 m/s and slowing down with a constant acceleration, I can calculate the time of travel.

Using my estimated acceleration from the bowling ball video along with an initial velocity of 6 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 direction (which it isn't). I can calculate the horizontal displacement using the basic kinematic equation (since the initial position is zero and the initial sideways velocity is zero):

Putting my values in, I get a sideways motion of 13 meters. That seems significant. But wait! This is for a ball going 6 m/s the whole time (even though I used a changing speed to calculate the time). I guess it could be significant if I did a more realistic calculation. Really, I should just do the numerical calculation of this.

Here's what I would love to see. First get a long East-West terminal and see if we can roll a ball all the way to the end of the hallway. There shouldn't be any Coriolis deflection in that case. Then take the same ball in a North-South terminal and see if there is a noticeable Coriolis deflection.

Maybe I should just carry around a bowling ball when I travel in case I see the perfect situation to test.

Homework: What would happen to this same problem on a smaller planet? How small would a planet have to be to have a very noticeable curvature in an airport terminal?