I love using seemingly random data to figure out stuff I wouldn’t otherwise know. You can do this with all kinds of things, but in this example, I use video that I recorded from an airplane to figure out how high and how fast it was traveling. Oh, and it explains why I like window seats for short flights.
Let me start with a frame from the video:
I filmed this while approaching New Orleans, so I know the approximate location. You can see it on Google Maps. No, I don’t know the exact location or altitude, but I know the angular size of objects in the video and the actual size of objects like roads and stuff from measurements on Google Maps. This is where knowing the most basic equation for angular size comes in handy. Suppose I have an object with a length L and a distance r from my camera. That gives me the following relationship (assuming L is much smaller than r):
Yes, this is essentially the same equation used to find the circumference of a circle if θ is measured in radians (which it should be). If you make θ equal to 2π, then the length is the same as the circumference. Of course, this means the object is not a straight line, but this equation still works fairly well with small angles.
I can determine the actual size of things using Google Maps, and I can use the video to measure their angular size. To do this, I must know the angular field of view for the camera. Good thing I already know this from an earlier experiment. Yes, that experiment used an iPhone 6, but I will assume the video camera on the iPhone 7 has the same horizontal angular field of view of 1.109 radians. To determine the actual angular size measurements, I will use Tracker Video Analysis—it works with videos and photographs.
Using the angular size to determine the distance to various objects as well as the actual distance along the ground, I can determine both the altitude and the true location. Let me explain with a diagram. Suppose the plane is at an altitude (h) and a distance (s) from a known point. After measuring the distance (r) and location of an object (x) on the ground, I get:
Since this is a right triangle, I can use the Pythagorean theorem to find a relationship between the three sides:
Remember, I don’t know h and I don’t know s, but I can find several values for r and x. So here’s the plan: Make a plot of r2 vs. x. It should be a parabolic equation. If I fit a parabola to this data, the coefficients should give me both h and s:
Technically, the coefficient in front of the x2 term should be 1.0, but I won’t worry about that right now. Instead, I will look at the coefficient in front of the x term. This should be equal to 2s and I get a fitting value of 4101.8 m. This means s should be half that value at 2050.9 m. I can use that to determine the exact location of the plane. What about the constant term from the fit? This should be equal to h2 such that the altitude of the plane is 3,283 meters.
Now that I know where the plane is, I can determine how fast it is moving. All I need to do is to track the motion of an object on the ground. Of course, I am seeing the angular motion of that object and not its speed—things that are farther away appear to move more slowly (this explains why the moon seems to follow you around). Tracking a point on the ground is like watching it move in a giant circle. If I measure the angular speed and I know the radius, I can find the true speed.
Here is a plot of the angular position of a point on the ground that is at a radius (from my previous analysis) of around 4,993 meters.
This is actually a plot of angle vs. time (not x). The slope of this line will give the angular velocity (ω)and I can use that with the following relationship:
With an angular velocity of 0.02328 radians per second, I get a ground speed of 116 m/s (260 mph). This means the plane is moving with the same velocity (but in the opposite direction). Yes, that seems a little slow, but it was during decent and probably higher than the stall speed. I think this value is OK.
But in the end, I calculated both the height and the speed of the aircraft based the video alone. Sure, there probably are better ways to do this, but what else are you going to do while waiing for your next flight?