This is one of my favorite stories. In short, one of John Burk’s (@occam98) students wanted to launch a space balloon. If you want all the details, this post at Quantum Progress pretty much says it all. The part that makes this story so cool is that it was the student who did all of the set up and fundraising and stuff. Love it. Oh, and the student is apparently named “M.” I wonder if the student is either one of the Men in Black or a James Bond scientist.
Ok, you know what I do, right? I need to add something. Here is a very nice video of the space balloon launch.
You know I like to use pictures for data from time to time, right? One problem is that I don’t know much about cameras. There, I said it. Really, almost all of my photos are made with my phone. That is what makes the phone so great, you almost always have your camera with you.
To make these pictures useful for physics, it helps to know the angular size of the picture. Here is a diagram so you can see what I am talking about:
There are 20 seconds left on the clock. Your team is down by 2 points such that a field goal would win it. The ball is spotted on the hash mark at the 15 yard line and it is first down. What to do? Should you call a run play so that the ball is in the center of the field? Or should the ball be kicked from where it is?
So there is the question. Is it better to kick the ball from an angle or move back and kick it head on? Let me just look at one aspect of this situation. What is the angular size of the goal post from the location of the kicker? I am not looking at the height of the horizontal goal post – I will assume the kicker can get the ball over this.
This was on reddit. It is an image from google maps showing an aircraft. Not surprising, there are lots of aircraft that get caught by the cameras in mid flight. But what about the colors? Is this some rainbow-unicorn plane? I am not sure of the exact details, but this rainbow effect is from the camera. I am not sure why, but this camera is capturing red green and blue (and probably white) colors separately at different times. Here is the actual link to the google map.
The first thing that comes to my mind is – I wonder how fast the plane was moving. That question is difficult to answer because I don’t know how much time was between each ‘color filter’ photo. Oh well, I will proceed anyway. First, some info. Reading through the very insightful reddit comments, it seems the commenters are certain that the plain is an Embraer ERJ 145. Really, all I need is the length. Wikipedia lists it with a 29.87 m length and a 20.04 meter wingspan. From the image, does the rainbow plane have the same ratio of length to wingspan as listed?
Ok, not quite the same. Maybe that is close enough. The one thing is that the image clearly has some distortion. Either the plane it turning or the image has been adjusted to make it look like it is a top down view. Well, surfing around a bit I couldn’t find another plane that was close in length/wing span ratio. I am going with ERJ 145.
If I scale the image from the length of the plane, how “far” between the different colors? Here is a plot of the 4 color images.
Note that for this image, I put the axis along the fuselage of the plane. The points are the locations of the back tip of one of the wings. The first cool thing that I can learn from this is that there must have been a cross-wind. The aircraft is not traveling in the direction that it is heading. Of course this is not uncommon, planes do this all the time. Oh, let me not that I am assume the aircraft is far enough away from the satellite that the multiple colors are due to the motion of the plane and not the satellite. This is probably a good assumption since the houses below are not rainbow colored.
What about the speed? If it is moving at a constant velocity, then:
I know the changes in position. So, let me just call the change in time 1 cs (cs for camera-second). This means that the plane’s speed would be 1.8 m/cs. Ok, let’s just play a game. What if the time between frames was 1/100th of a second? That would mean that the speed would be 180 m/s or 400 mph. That is possible since wikipedia lists the max speed at around 550 mph. If the time between images is 1/30th of a second (I picked that because that is a common frame rate for video) then the speed would be 54 m/s (120 mph). That doesn’t seem too low. I would imaging the landing speed would be around that speed (or maybe a little lower – but what do I know?)
But WAIT – there is more. Can I determine the altitude of the plane? Well, suppose I have two objects of two different lengths that are two different distances from a camera. Here is an example.
My notation here looks a little messy, but both objects have a length (L) and a distance from the camera (r). They also have an angular size, denoted by θ. About angular size, I can write the following.
I don’t know the distances from the camera and I don’t know the angles. But, I can sort of measure the angles. Suppose I measure the number of pixels each object takes up in the photo. Then the angular size could be written as:
Where p1 is the pixel size of an object and c is some constant for that particular camera. Now I can re-write these angular equations and divide so that I get rid of the c.
I can get values for all the stuff on the right of that equation. Here are my values (object 1 is the plane and object 2 is the background – really, I will just use the scale provided by google maps). Oh, one more thing. I am not going to measure the pixel length but rather some arbitrary length of the same scale.
L1 = 29.87 m
p1 = 1 unit
L2 = 10 m
p2 = 0.239 unit
Putting in my values above I get the ratio of the distances from the camera as:
Now I just need one of the r‘s – ideally it would be r2 (the distance the camera is from the ground). Wikipedia says that the satellite images are typically taken from an aircraft flying 800-1500 feet high. So, suppose r2 = 1500 feet (457 meters). In this case the altitude of the rainbow plane would be:
1000 feet would mean that the rainbow plane is probably landing (or taking off). It looks like Teterboro Airport is quite close and the rainbow plane is heading that way. I claim landing.
So, here is what I can say:
Airspeed. Really, I don’t have a definite answer. Like I said before it depends on the camera rate. If I had to pick (and I don’t) I would say that the rainbow plane is going 120 mph and the time between different colored images is 1/30th of a second.
Altitude. If I go with the higher value of the typical google-map planes (like the google map cars but with wings) then the altitude would be around 1000 feet. This lower altitude is why I used the lower value for the airspeed.
Windspeed. Now I am changing my answer for windspeed. I am going to pretend like there is no wind. The perpendicular motion of the colored images could be due to the motion of the google-map plane.