Rocks and Canoes

This is an attempted proof for the rock tossed overboard problem and what happens to the water level.

"You sit in a canoe with a rock.  You throw it overboard, and it sinks.  Does the rock make the water level go up, down, or stay the same when thrown in?"

Does the water level go up?

Archimedes principle states that when a body is immersed in a fluid, the fluid exerts an upward force on the body equal to the weight of the fluid the body displaces.

Since the rock doesn't sink the canoe, its weight or downward force is being equaled by the buoyancy of the water, an upward force.

B = pgV

B - buoyancy force
p - density of water
g - gravity
V - volume of water displaced

So the weight of the rock is equal to B

w = mg

w - weight or downward force
m - mass of rock
g - gravity

So

B = w
pgV = mg

When the rock is thrown overboard, the rock sinks; it overcomes the buoyant force of water.

In canoe
pgV = mg
In water
pgV < mg

Now 'p', 'g', and 'm' do not change.  That being, the density of water, the gravity, and the mass of the rock all stay the same.  The only thing left to change is the volume of water displaced, 'V'.  The volume of water displaced must go down, and so the water level must go down.

Could the water level stay the same?

The problem with this is the rock must sink as set by the question.  To keep the water level the same the rock would need to be equal in density to the water.  Water doesn't sink in water, and neither would an object of equal density.

m = pV
m - mass
p - density
V - volume

You can substitute pV for the rock's mass

Before
water         rock
pgV      =   pgV
After
pgV      =   pgV

If the rock was going to sink the downward force would have to be greater.  The left side doesn't change, and volume and gravity on the right don't, so the rock's density doesn't.  Your force is still equal; the rock wouldn't sink.

However you look at it the water level goes down.