Understanding the Relationship Between Dripping Rate and Water Clock Design

[That Neuron]

Okay, this is a really simple question, so to anyone looking for some extraordinarily complex differential equation question turn away now, or be blinded by boredom.

My query is rooted in a question I had about building a water clock... so seemingly relevant to Differentials, I know. Anyways, I realized that the rate of dripping (though probably much more complex than a proportionality) was at simplest proportional to the height, or at least related to it.

Anyway, I was thinking that if the rate of change of the height is proportional to the pressure on the hole at the bottom out of which water drips (or pours) then I could create the differential dy/dt = -k (πr² pg y(t), where p is equal to the density and g is the acceleration due to gravity, this equation translates to y = y(0) e^-kπr2pgt.

But this function seems to decline too steeply for this application, am I doing this right?

 

[MostlyHarmless]

Okay, this is a really simple question, so to anyone looking for some extraordinarily complex differential equation question turn away now, or be blinded by boredom.

My query is rooted in a question I had about building a water clock... so seemingly relevant to Differentials, I know. Anyways, I realized that the rate of dripping (though probably much more complex than a proportionality) was at simplest proportional to the height, or at least related to it.

Anyway, I was thinking that if the rate of change of the height is proportional to the pressure on the hole at the bottom out of which water drips (or pours) then I could create the differential dy/dt = -k (πr² pg y(t), where p is equal to the density and g is the acceleration due to gravity, this equation translates to y = y(0) e^-kπr2pgt.

But this function seems to decline too steeply for this application, am I doing this right?

Is you're equation dy/dt=-k(...pg)y(t) where "y(t)" is y as a function of t. or the variable y times the variable t?

 

[That Neuron]

Oh, no y is a function of t!

Not the height!

I actually think that I found the correct function simply by playing around with the constant k. Let me revisit this!