Blog du Wren

Mathematical Mythbusting

Two occasions recently have had me pondering the speed of sound. The first such occasion was a lightening storm, of which we have very few here in the rain shadow of the Cascades. Lightening and thunder filled the sky all evening and into the night. Secondly, as I mentioned in my previous post, I watched the fireworks from a significant distance, which had an influence not only on the volume of the bang, but also on the delay.

In talking to people (esp. about the lightening, although I don't see what it wouldn't also apply to the fireworks), the rule of 1 second between sight and sound equals one mile of distance to said sight was oft cited. I myself used this rule in my early days. This time around, however, I got a little curious about that rule of thumb and wanted to run the numbers. Perhaps to your surprise, the rule isn't so accurate. Observe*:

Givens
Speed of Sound	S_s = 761 mph (or .2 miles per second)
Speed of Light	S_c = 670,616,629 mph (or 186,282 miles per second)
System of equations
Time it takes light to reach you	t_c = d/S_c
Time it takes sound to reach you	t_s = d/S_s
Time elapsed between light and sound	t_e = t_s -t_c
Solving
Solving for t_c	t_c = t_s - t_e
Replacing for t_c and t_s	d/S_s - t_e = d/S_c
Moving stuff around	d/S_s - d/S_c = t_e
Factoring for d	d(1/S_s - 1/S_c) = t_e
Solve for d	d = t_e / (1/S_s - 1/S_c)
Plugging givens back in (using miles per second)	d = t_e / (5 - .000001*)

This shows that instead of every one second between lightening and thunder equating to one mile, every 5 seconds equals to a mile. So next time you hear that thunder 5 seconds after the lightening, don't go rushing outside thinking you've got a 5 mile buffer because it's a lot closer than you think.

Thursday July 10 2008

File under: misc

Toggle Comments (6)

comment?

on Thu 10th Jul, 2008 10:46 am PDT Saxtor said:
As with all equations like this, it is also the assumption that it is happening in a vacuum. I would also guess that anomalies such as winds, obstacles and Jessica Lynch of Slow Loris Shirts could also cause delay variables.
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on Thu 10th Jul, 2008 10:52 am PDT Wren said:
Actually, Saxtor, sound doesn't propagate in a vacuum*, although winds and obstacles are sure to make a difference.
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on Fri 11th Jul, 2008 12:25 am PDT Horge said:
Seems logical to me. I wonder if there are any other formulas out there that could be figured out to produce (and spread) more common sense rules of thumb. Like, how far away does a cat have to be for a dog to start barking at it? I imagine it's a bit more complex, involving the speed of light (vision), the speed of sound (the resulting barking, or yipping if you have a chihuahua) and the speed of smell. I think that studies of the speed of smell are grossly understudied...
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on Fri 11th Jul, 2008 07:50 am PDT Sean said:
I've never heard of that rule and I'm glad, cause whoever started spreading that should be kicked in the junk.Now what about the speed of gravity? If the sun exploded we'd instantaneously feel the effects, ja?
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on Fri 11th Jul, 2008 09:52 am PDT Sweet Julie said:
Ooh, Ooh! Can you do a post about the Golden Number aka Fibonacci Numbers? I love geeky science stuff! :) (just a reminder- voting is still going on for the Gray Whale Fluke!)
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on Mon 14th Jul, 2008 09:51 pm PDT Ryan Walters said:
Sean--no, gravity propagates at the speed of light, so if the sun were to suddenly disappear we wouldn't notice for about eight minutes.Wren--Anacortes is in the rain shadow of the Olympics, not the Cascades.
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