9-March-2015 Modeling Air Resistance

Purpose: Determining the relationship between air resistance force and speed, and seeing if we can model the tall of an object including air resistance.

Theory: Once an object reaches terminal velocity while falling, the force of friction from the air and the weight of the object are the net force, which is equal to the mass and its acceleration. We can use this relation to create an expression for air resistance with velocity as a term, and can test how accurate our model is by comparing the predicted velocity vs. actual measured velocity. Hopefully with only 10% of error.

Procedure: Take five coffee filters, weigh them to determine mass, drop one from a certain height, record the fall, place another coffee filter atop the previous one, and repeat dropping and recording until you have used all five filters. Load videos into logger pro, in order to determine velocity of each trial run, record velocity, create a velocity vs mass times gravity, use it to create an equation/model for air resistance, test model on excel and compare it to the recorded velocity to determine how accurate it was.

*First we dropped a coffee filter from a balcony.

*Loading the video into logger pro, we were able to determine the velocity by plotting points at each frame of where in relation to the balcony (roughly 1 meter) the filter was while falling.

*We repeated the process, always adding another coffee filter to increase the mass, five times.

*We determined the terminal velocity of each run, recorded the mass of each run, and calculated the force of mass times acceleration due to gravity of each run.

Recordings

V (m/s)	M (kg)	F (N)
0.9578	0.0008947	0.008768
1.323	0.001758	0.01723
1.605	0.002684	0.0263
1.922	0.003579	0.03507
2.253	0.004473	0.04384

*We created a velocity vs force of mass*gravity graph, since the only two forces acting on the falling filter are air resistance (with velocity as a factor in the equation) and the filter mass times acceleration due to gravity, we hope to see some type of relation between the two on a graph.

Graph

*The graph is exponential, and when fitted to give an equation it gives us the force of the mass times gravity is equal to the velocity times a constant taken to a power (mg=Av^B).

*A and B are constants, and represent factors that would effect air resistance, like surface area and shape and friction of the air.

*The value for A given from logger pro is 0.01105 and B is 1.725, and v is the terminal velocity of a falling coffee filter

*We now test our model using excel.

*Our model is the net force(the mass as it accelerates downwards) will equal the force of weight minus the force of air resistance

                                                                  (ma=mg-Av^B)

*In time increments of only .001 seconds, the change in velocity will be the acceleration determined from our model multiplied by the time increment.

*Eventually the changes in velocity will become so minute, the velocity will be virtually unchanged by them, this is our terminal velocity.

*For a single filter of mass 0.0008947kg

Results

t=	0.001
A=	0.01105
B=	1.725
m=	0.0008947
t (s)	a (m/s^2)	∆v (m/s)	v (m/s)	∆x (m)	x (m)
0	9.8	0	0	0	0
0.001	9.79576796	0.0098	0.0098	0.0000098	0.0000098

*Scrolling down

t (s)	a (m/s^2)	∆v (m/s)	v (m/s)	∆x (m)	x (m)
0.485	0.00216469	2.2074E-06	0.8743945	0.00087439	0.36720737
0.486	0.00212285	2.1647E-06	0.87439666	0.0008744	0.36808177
0.487	0.00208182	2.1229E-06	0.87439879	0.0008744	0.36895617
0.488	0.00204158	2.0818E-06	0.87440087	0.0008744	0.36983057
0.489	0.00200212	2.0416E-06	0.87440291	0.0008744	0.37070497
0.49	0.00196342	2.0021E-06	0.87440491	0.0008744	0.37157938

*At close to 0.486 of second the velocity to four significant figures "settles" at 0.8744 m/s, all further changes to velocity will be so miniscule, it can be argued that the filter is now at terminal velocity.

*For the first run the "predicted" velocity was 0.8744 m/s, actual 0.9578, error 8.7%

*For the first run the "predicted" velocity was 1.304 m/s, actual 1.323, error 1.4%

*For the first run the "predicted" velocity was 1.653 m/s, actual 1.605, error 2.9%

*For the first run the "predicted" velocity was 1.953 m/s, actual 1.922, error 1.6%

*For the first run the "predicted" velocity was 2.222 m/s, actual 2.253, error 1.4%

Conclusion

*My model worked very well, all "predicted" terminal velocities were only off by less than 10%, so I am satisfied with the values of A and B, and the formula Av^B as a model for the force of air resistance.

*Perhaps the greater error for the first run was because of how more "malleable" a single coffee filter is, its surface area and shape could change easier while it fell vs. multiple coffee filters holding each others shape in place.

*The whole process might be time consuming, as the constants A and B change depending on the shape of the object, its surface area, and air temperature effecting air density. All contribute to air resistance, so the entire experiment would have to be done again for another object of differing shape and multiple outside temperature.

2-March-2015 Nonconsant Acceleration Problem

Purpose: Because of how time consuming solving a integral can be, if it can be done at all, when calculating something (like acceleration) that is not always constant, it is sometimes best to solve it numerically using the raw computing power of a machine.

Theory: That if the time integral is small enough while using excel, we can determine how far the hypothetical 5000kg elephant, on frictionless roller skates, going 25m/s at first, then a rocket on its back generates 8000N thrust opposite its current motion, and the rockets mass changes over time and can be expressed m(t)=1500kg-20kg/s*t, numerically rather than analytically.

Procedure: Inputting formulas into excel

*First we set up the basic acceleration equation (acceleration=force/mass) with time as the only variable, given the situation. The elephant has a -8000N force heading in the opposite original direction and the mass is 5000kg(elephant) plus a rocket of 1500kg which is losing mass at 20kg a second.


                  Acceleration=-8000N/(6500kg-20kg*Time) or reduced too a= -400/(325-t)

*We input this formula into one of the cells, scroll down a satisfactory length, here about 220 rows.

*For time we're going to do increments of only .1 seconds, so we create a column that constantly increases by only .1 seconds, scroll down 220 rows, and have that time be placed into the adjacent cell for calculating acceleration.

*In another column we can now take the average acceleration of each preceding two calculated accelerations and divide by two.

*If we multiply the average acceleration by the each increment of time (0.1 seconds), we have calculated the change in velocity, which since the elephant is slowing down should be negative. Scroll down to make a column

*Adding the change in velocity (really how much the velocity is decreasing by) to the original results in the velocity at the specific time. Scroll down to make a column.

*With a column of velocity we can create an average velocity at each moment, using the same method as for average acceleration. Scroll down to create a column

*With a column of average velocity we can determine the change in distance by multiplying by the 0.1 second time increment for each cell. Scroll down to create a column

* Adding each increment of change in distance for a total distance travelled.

*If everything was done correctly the first few rows should look like this

Results

T s	a m/s^2	a_avg m/s^2	∆v m/s	v m/s	v_avg m/s	∆x m	x m
0	-1.23076	0		25			0
0.1	-1.2311	-1.23095	-0.12309	24.87690	24.93845	2.493845	2.493845
0.2	-1.23152	-1.23133	-0.12313	24.75377	24.81533	2.481533	4.975378
0.3	-1.23190	-1.23171	-0.12317	24.63059	24.69218	2.469218	7.444597

*If we scroll down, eventually the increments in distance start to decrease instead of decrease, because the average velocity is becoming negative, meaning the elephant is finally moving in the opposite direction around the 19.7 second mark.

Results

19.6	-1.30975	-1.30954	-0.13095	0.118883	0.184360	0.018436	248.692701
19.7	-1.31018	-1.30997	-0.13099	-0.01211	0.053385	0.005338	248.69804
19.8	-1.3106	-1.31040	-0.13104	-0.14315	-0.07763	-0.00776	248.690276

*Looking at the chart this is the furthest the elephant goes 248.7 meters, and solving the original equation for acceleration through integration is also 248.7 meters.

Conclusion

*In comparison, the numerical method is a valid form to determine an answer over the analytical method, as long as you have a calculator that can do all the blunt force calculations. As excel has determined the distance traveled, just as hand written calculus would.

*The smaller the integral of time the smaller the change in calculating the distance at each point in time, too large could result in a few meters changing in each cell vs the 1/100th of change. So the integral would be "small enough" depending on how many significant figures you need.

*You know if you've reached the max point of distance travelled when the distance is starting to decrease, meaning the elephants velocity is finally heading in a different direction.

4-March-2015 Propagated Uncertanity in Measuerments

Purpose: An introduction to error calculations, using known density of metals and comparing them in lab measurements. Since the metals density are known, any deviation when checked with our measurements must be do to the accumulated error, for part one. Part two will be determining the mass of two hanging unknown masses, using force gages and the uncertainty in them and the angles to determine the accumulated error.

Theory: My measurements will be close, 10%, to the expected value, and any error will be predicted from the predicted errors that come with certainty in measurement.

Procedure: We will take three metal cylinders of different types (copper, iron, and aluminum) and sizes, measure their mass, diameter, and height with calipers. Calculate the mass and then calculate the propagated error to determine if they are within the experimental uncertainty.

* We determined the mass, height, and diameter of all three metal cylinders, all measurements with a +/- 0.05 level of certainty.

* We calculated the volume and densities

*Determined that the partial derivative of volume with regards to mass as 4/(Pi*h*d^2)

*Determined that the partial derivative of volume with regards to diameter as -8m/(Pi*h*d^3)

*Determined that the partial derivative of volume with regards to height as -4m/(Pi*h^2*d^2)

*The total uncertainty in the density of each cylinder would be determined by adding all the partial derivatives, multiplied by the original uncertainty (P=partial derivative mass *(uncertainty)+ partial derivative of height*(uncertainty)+ partial derivative of the diameter*(uncertainty)

*For iron we got 6.68 g/cm^3 +/-  0.13, and actual 7.87 g/cm^3, error 13%

*For copper we got 8.94 g/cm^3 +/-  0.18 and the actual  8.96 g/cm^3, error 1.8%

*For Aluminum we got 2.81 g/cm^3 +/- 0.05 and the actual was 2.7 g/cm^3, error 6%

Conclusion

* All but one of the metals fell into my expected error, iron.

*I still believe this is a good method for determining density, as the other two metals were very accurate to the expected density.

*Perhaps my sample wasn't pure iron, as checked the periodic table and no element has a similar density to the one I calculated.

Part 2
Theory: The accumulated error will be accounted for in my calculations.

Procedure: Take measurements of two hanging masses, the angles and force gages, and calculate the propagated error for each

Recordings of Mass 1
 

Mass 1	Unknown
Force Gage 1	5.75
Uncertainty	0.1
Angle 1	8.5
Uncertainty	1
Force Gage 2	9.5
Uncertainty	0.3
Angle 2	54
Uncertainty	1

 *Calculating mass with uncertainty 0.871kg +/- 0.0355

Recordings of Mass 2

Mass 2	Unknown
Force Gage 1	7
Uncertainty	0.1
Angle 1	27
Uncertainty	1
Force Gage 2	7.3
Uncertainty	0.3
Angle 2	40
Uncertainty	1

*Calculating mass with uncertainty 0.8031 +/- 0.0454