Why does Toricelli's Principle result in an exponential function for h vs. t?

[Muzly]

OK - I'm working on Toricelli's Principle at the moment. That is, a tank where there's a small hole at the bottom and we're supposed to plot h (height) vs. t (time), V_jet vs. t, mass flow rate vs. t, and change in height vs. t.

Now - I get a linear graph for V_jet and mass flow rate vs. time, but a non-linear curve for h and mass in the tank vs time. And I can't prove it.

We're neglecting any internal energy transfers, etc. just plain old V = Sqrt(2.g.h)

Any help muchly appreciated.

 

[HallsofIvy]

If "mass flow rate" vs time is linear, rate= kt, then mass moved out of the tank will be quadratice in time: (1/2)kt²; so of course, the mass in the tank will be quadratic.

 

[pixel01]

Let mass vs t is v(kg/min) so : h' =-A*v( A is a constant)
And we can see that v is linearly propotional to h or v=B*h
then combine the two we have:
h'=-C*h
Solve this differential equation, we have the function h=D*exp(-k.t). So h=f(t) is not a quadratic, it's an exponential.