Thursday, March 4, 2010

Real World Related Rates

Final Project Group 1
Related Rates
Co-Authors: Blitzen, Dammitimmad, and mc Casper.

Materials:
Bowling Ball

Marble Video
Camera
Smooth floor (with minimal friction)
Paper and pen to do the work
Black Box (calculator)
Grey Box (brain)

Purpose: For our final project we were asked to create a related rates problem within the real world using a program to draw the actual data from our experiment. Using the video analysis with a program called logger pro we had all of the data needed to solve the problem at hand.

The Problem: Our experiment consisted of rolling a b
owling ball and a marble away from each other at a 90 degree angle. We executed this experiment on a smooth and level concrete floor indoors. We can assume that both balls were rolling at approximately the same speed (see next paragraph below). Our goal is to find the rate of change of that the distance between the two balls is increasing with respect to time.

The great lengths of assuring a constant velocity:
Time frame of experiment was roughly 2 seconds
Very smooth concrete floor used in experiment
No interference after the initial push off
Air resistance is negligible due to being indoors and
no wind

Figure 1:
Video of actual experiment!
video


Data:

Graph of our data:
You can see from our graph above that the velocity of the two objects is within minor human error, constant. The only part of the lines that are not constant is the beginning portion of the graph due to the ball's being stationary at the beginning of the video. The slight curve that is also apparent before the constant velocity is reached is due to the initial push from us accelerating the balls and bringing them up to speed.

Using the linear regression on our program (called logger pro), we got the equations for the velocity and displacement of each object individually.

Graphs with the linear regression equations.
(Because the linear equations are difficult to read, you can go to our WikiWork Problem to see the equations of these lines more clearly.)
http://crmscalc2010.pbworks.com/WikiWork-Problem-2

General Overview How To Solve:
After obtaining the data and the linear regressions in loggerpro, we found the velocities using the data begotten from loggerpro. We used the X and Y components of object's velocity to find their total velocity.
Next we chose a random point somewhere from the middle of the graph.
To find the rate of change of the distance between the two objects we used the derivative of Pythagorean formula (see wiki). We substituted in the displacements for both objects into the formula and their velocities to find dz/dt (the related rate of the distance between the two objects with respect to time).

We choose to repeat this process for three separate points to see how accurate and consistent our answers were.

After completeing our project we determined that there was very little human error in our experiment because all three final answers were only one tenth different from one another.
***See WikiWork problem for full explination of calculations.***

Final Problem (one last time):
What is the rate at which the distance between the two objects is increasing? The velocity of the bowling ball is 37.957 inches per second. The velocity of the marble is 43.52 inches per second.

***Please see our WikiWork page to see the calculations and solution to this problem***
http://crmscalc2010.pbworks.com/WikiWork-Problem-2

1 comment:

  1. Hey Linnea,
    Great start. I think this looks fantastic and you rock. One thing that you might want to include is just a very general outline of the calculations (ie, we found the component displacements in loggerpro, found the component velocities using loggerpro, combined the component velocities to get total velocity, then took random points along the trajectory to determine the rate of change of the distance between the two objects, used the displacement values to determine the distances, then used the distances and the previously discovered velocities to find the rate of change) or something like that.

    Also, when you describe the experiment, they weren't rolled away from eachother at 90 degrees. although, thanks to loggerpro, we are able to add the components of each in order to make the trajectories 90 degrees. But that sounds wrong to me, IDK, maybe include that, but its late and i'm tired so i could just be stupid right now, so you might not want to include that. actually, that probably doesn't make sense, so maybe don't include it.

    but other than that, great, (and yes, i know, i can sound stupid sometimes)

    cool

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