Air and Water Project using MATLAB, (DVQ Math).

/in /by

Air and Water Project using MATLAB, (DVQ Math).

The goals for this project include one or more of the following:

  1. Using MATLAB with the dsolve and ezplot commands
  2. Making use of script M-files in MATLAB
  3. Using MATLAB along with if/then statements to perform approximation techniques, and bring in the plot command; or do the aforementioned in a spreadsheet (most likely Excel)
  4. In either a spreadsheet or with MATLAB, approximate a system of differential equations by tweaking previous files to adapt to the new problem.

The Project is:

When velocities are high, experiments determine that the model we’ve used

m:frac{dv}{dt}=:mg:-:k_av where down is positive and up is negative

where kk is some constant of proportionality does not adequately model what occurs:  experimental data does not fit the model, unless vv isn’t terribly large.

This is especially true when moving through water, where even slow velocities are shown to have a disproportional effect on the drag.  In fact, it increases almost proportional to the square of the velocity:

m frac {dv}{dt} = mg - k_wv^2mdvdt=mg−kwv2where down is positive and up is negative, and motion is always downward.  

In addition, when moving through water there is often a buoyancy force due to the displacement of water; we should actually use 

m frac {dv}{dt} = mg - F_B - k_wv^2mdvdt=mg−FB−kwv2 where F_BFB is the buoyancy force.

On to the problem:

Suppose an object with a mass of 30 kg is dropped from the edge of a quarry pit and falls down to the bottom, where water has collected.  We know the water is 3 m deep.  

Due to prior experiments,we have determined that the object has a terminal velocity of 15 m/s in the air, and only 2 m/s in water (it falls slowly in water) due to water resistance and a buoyancy of 200 N.  

Listening for a splash, we know it hits the water after falling for 4.5 seconds.  

1.  What are k_aka and k_wkw equal to?  In order to give us the observed terminal velocities in air and water, we should be able to calculate them.  (Probably something to do without MATLAB.)

2.  How fast is the object traveling when it hits the water’s surface?  (Use MATLAB to solve for velocity in the air.)

3.  What distance did it fall to the surface of the water from the release point?  (Use MATLAB to solve for position in the air.)

3.  How long after its release will it reach the underwater bottom of the quarry?  (Use MATLAB to solve for position in the water.)

4.  How fast will the object be traveling when it hits the bottom?  (Use MATLAB to solve for velocity in the water.)

Aside from the dsolve command, you might want to use graphs generated by ezplot to estimate answers, or write a root finder; you might also try using the solve command, but it might not give you the results you expect.

-Solve this problem using MATLAB; discuss your observations.  Include any graphs or other helpful information (screen shots, or the text of what you typed in, or any script M-files) should be attached; it’s preferable to have all of this in one document, either a *.doc, *.docx, or *.pdf.