Use of separation of variables and graphical analysis in solving a special case of Torricelli’s Law
 
 
 
 
 
 

In this problem we are presented with a situation where we have a can full of water with a small hole in the bottom. We need to know how long it will take a certain amount of water to drain out of the hole. By use of a special case of Torricelli’s Law, we observe that the change in the height of water in the can is proportional to the square root of the height, that is (dy/dx) = k (y)^1/2. By use of this law, we can determine the amount of time required for the water to drain.

Given the following set of data, we will apply Torricelli’s law and use the separation of variable technique to solve this differential equation.

Original Data
 

t (minutes)	0	0.5	1	1.5	2
Y (inches)	3.063	2.5	2.125	1.75	1.375

 

Below is a graph of the original data points:

 Original Data Graph
 
 

First, we need to solve the differential equation:

(1)  (dy/dt) = k (y)^1/2
 
 
 

We need to separate the variables to get:

(2)  1/(y)^1/2 (dy) = k (dt)
 
 
 
 

We then integrate to get:

(3)     y^1/2
         --------   = kt+C
           1/2
 
 

This is simplified to get:

(4)   2 (Y)^1/2 = kt + C
 
 

We now need to find a linear relationship between the variables in order to produce a graph of the data.

If we let Y = 2(y)^1/2 and let T = (t), we can rewrite the equation as:
 

(5)  Y = kT + C
 

We can now rewrite the data as:

Transformed Data
 

T (minutes)	0	0.5	1	1.5	2
Y (inches)	3.5	3.162	2.915	2.646	2.345

 

We now are able to plot these points and using a HP Calculator, we can build a Linear Regression Graph, which will give us the values for C and for k.
 
 
 

We use the Slope of this line to find k and we use the Y-Intercept as the value of C.

k = -.5652

C = 3.4788
 
 

With this information in hand, we can now rewrite our equation as:

(6)  Y = -. 5652 T + 3.4788
 

 Linear Regression Graph
 
 

By substituting Y with 2*(Y)^1/2, our equation now reads

(7)   2* (Y)^1/2 = -. 5652 T + 3.4788
 

In order to determine the time (T) where all of the liquid has drained, we can set (Y) = 0 and we get:
 

(8)   2 * (0)^1/2 = -. 5652 T + 3.4788
 

By solving for (T), we find that (T) = 6.155 sec. This is the amount of time it takes the liquid to drain from the can.
 

Now let us look for the best fitting curve. We start with the equation:

(9)   2* (Y)^1/2 = -. 5652 T + 3.4788
 

Let’s solve for Y. we get:

(10)   Y = (1.7394 - .2825 T)^2.
 

Below is the graph of this curve:
 

 Estimated Graph
 
 
 

By taking our original data, we transformed the data and found a linear relationship

between time T and the change in height Y. We were then able to compute our constant

of proportionality k and determine the constant C. Using this equation, we were able to

determine the amount of time T that it took for the can of soup to drain completely. Then

we took the information from the linear graph and constructed an estimated function

curve that fit the original data points. This estimated function models the original data.