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 = 16t^{2}
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 = 16t^{2}
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. |