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Approximating The Depth of a Lake Using Mathematics

Introduction - A Lake With Fish As Big As Cars!
While visiting northwest Arkansas a number of years back, I was told by some local residents that Beaver Lake, an impoundment of the White River, contained catfish "as big as small cars".  I reasoned that Beaver Lake must be a very large and deep lake to contain large fish, although I never bought into the idea that the fish were actually as big as cars.

Calculating The Depth Experimentally
We happened to be sightseeing at the location of the Beaver Lake Dam which happens to have a road built on the top of the dam.  I thought "I can check the depth here with a simple experiment!"  I remembered from Physics that a projectile launched horizontally takes the same amount of time to fall as one that is simply dropped.  Also, I remembered that the freefall equation is S = 16t2 where S represents feet of drop and t represents time in seconds for an object dropped with negligible air resistance.  I needed to throw a rock straight out since I could not get to the edge of the dam to drop it.  After throwing the rock straight out, I found that it took about 4 seconds for the rock to hit the water below.  Plugging t=4 into S = 16t2 results in 256 feet.  Since the depth of the lake will be at least as much as the height of the dam, I calculated that the lake was around 250 ft deep on the reservoir side!

The Actual Depth - I Was Close!
After inquiring with the US Army Corps of Engineers,  I received the following email:

The estimation method worked quite well, considering I had no stopwatch and I estimated the seconds by roughly counting them off. The giant catfish rumor proved to be just that. A rumor.

Online Projectile Experiments
Check out This Free Fall and Projectile Simulator.  You can verify that a projectile thrown straight out takes the same amount of time to hit the ground as one that is dropped.

 

 

 

 

 

 

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