Zamboni!!

Last week's question: How can you estimate the speed of a passing plane? Certain assumptions are allowed. In the first case, let the distances sound and light traveled be the same. In the diagram and equations below, the variables are

• xl - the distance traveled by the plane since it produced the sound you're hearing. l is the length of the plane, which you can look up by the model. x is how many plane lengths there are between the source of sound and the source of light.
• theta - the angle between the paths of light and sound.
  
• d - the distance traveled by light and sound. d_light = d_sound
• theta_s, theta_l - the angle opposite the paths of sound and of light, respectively.
• v_? - the velocity of various entities. The speed of sound and light are given.
  

To get v_plane, first we find d through the law of sines:

Then we equate the travel times of light, sound, and the plane. Assume the plane has moved a negligible amount between the reflection of light to your sighting of the plane. This gives:

That was pretty easy, wasn't it? This formula seems roughly correct; if the sound and sight of the plane forms a 30 degree angle, the planeis traveling at 176 m/s. Online sources say a B747 actually travelsat around 240 m/s, which would be about 41 degrees.

Now for the bonus part: what if the distance traveled by light and sound were not equal? Then we have the situation below:

I don't actually have a neat equation for this one, but I managed to set up four (hopefully) independent equations, which will allow us to solve this system. The four equations are below.

The simple ones first. By the sum of angles in a triangle:

By the same relation, and the trigonometric tangent relationship:

By the law of sines:

Finally, this mess is created by using Heron's formula and the trigonometric sine relationship:

That gives us four equations in the four variables d_light, d_sound, theta_l, and theta_s. As I said, these should be independent; if someone solves this (or fails), please let me know.

After these four are solved for, we use the same time equivalence equation:

And we're done.

Note that in both cases, if you are also given the angle the plane and the horizon phi, you can also calculate the height of the plane:

Faye tagged me for a photo thing, and after questioning her definitions, I have this:


This was taken while I was at CTY. We were walking back to Stanford from dinner, and passed by the Facebook headquarters. From left to right, we have Sam, Lizzy, Hina, Erin, and me; Erica's behind the camera. We're all TAs for different courses. You can see more of this set on Picasa, and the tagged version on Facebook.

To compensate for the tough question, this week's will be merely fact finding, both related to ice rinks:

• Are outdoor ice rinks actually made from pure water?
• Is Zamboni a brand, or the name of the actual machine?