Torricelli’s Law If we punch a hole in a bucket full of water, the fluid drains at a ran...

Torricelli’s Law If we punch a hole in a bucket full of water, the fluid drains at a range governed by Torricelli’s law, which states that the rate of change of volume is proportional to the square root of the height of the fluid

The rate equation given in Figure 3.2.11 arises from Bernoulli's principle in fluid dynamics, which states that the quantity is constant. Here P is pressure, is fluid density, v is velocity, and g is the acceleration due to gravity. Comparing the top of the fluid, at the height h, to the fluid at the hole, we have

If the pressure at the top and the pressure at the bottom are both atmospheric pressure and if the drainage hole radius is much less than the radius of the bucket, then

leads to Torricelli's law: we have the differential equation

In this problem, we seek a comparison of Torricelli’s differential equation with actual data

(a) If the water is at a height h, we can find the volume of water in the bucket by the formula

In which denote the top and bottom radii of the bucket, respectively , and H denoted the height of the bucket. Taking this formula as given, differentiate to find a relationship between the rates dV/dt and dh/dt.

(b) Use the relationship derived in part(a) to find a differential equation h(t) (that is, you should have an independent variable t, a dependent variable h, and constants in the equation

(c) Solve this differential equation using separation of variables. It is relatively straightforward to determine time as a function of height, but solving for height as a function of time may be difficult.

(d) Obtain a flowerpot, fill it with water, and watch it drain. At a fixed set of heights, record the time at which the water reaches the height .Compare the results to the differential equation’s solution

(e) It has been observed that a more accurate differential equation is

Solve this differential equation and compare to the results of part (d).