The purpose of this lab is twofold. Firstly to determine a relationship between an objects terminal velocity and its mass, and secondly to model an object falling with air resistance accounted for.
In order to do this, me and my group needed to collect data from falling objects, and because we couldn't attach a spark tape or similar measuring device to our falling object without affecting our results due to extra mass, air resistance and tension, we had to use a camera to capture the motion of our falling objects, which in this case were coffee filters. These coffee filters to be exact:
The process of data collection was as follows, coffee filters were dropped, first one at a time, then two stacked together, then three...ect until 5 coffee filters (stacked together to make one unified mass with roughly the same amount of air resistance as 1 coffee filter alone) were dropped off of this balcony:
A meter-stick was put against the glass wall on the balcony to allow for scaling when the data was analyzed. Each drop was filmed from the moment the filters were released to the moment they hit the ground.
NOTE: I would like to now post many more pictures of our data collection process, but unfortunately all of our video footage was lost after we obtained all the relevant data from them.
Having obtained 4 videos for analysis, my group returned to the lab in order to get useful data out of them. To do so, we used loggerpro to plot a position to time graph based on the motion of the coffee filters on our captured film. In all films, the slope of that graph eventually became relitivly constant, which of course meant that velocity is increasing at a constant rate, meaning that net force on the coffee filters had reached 0.
This was expected, because the force of air resistance on an object is related to its speed by the relationship:
And net force on the object will be equal to (assuming down to be the positive direction for x) mg-kv^n. At some point then, the object will reach a speed v which makes the force kv^n and mg equal and opposite. This value is referred to as the terminal velocity.
All of this gives us two useful bits of information, first, the force of the air resistance in each trail at terminal velocity, which is equal to mg (since Fnet=0=mg-kv^n) and the terminal velocity itself. We can (and did) plot force as a function of terminal velocity and achive a graph which should obey the power rule above (that is Fresistance=kv^n). This graph could then be analyzed by logger pro for a curve fit, giving us values for k and n, the graph and these values are as follows:
As we can see, from this we get the values of k=0.01092 +/- 0.0007425 and n=2.162 +/- 0.1645, Giving us the equation:
Fresistance=0.01092(v)^2.162
in order to test these values, we set up a graph of one of our trials (the 4th trail to be exact) and our model of how that coffee filter should have fallen based off of our new equation. To set up our equation in excel, we did the following calculations:
Which led to the following data set:
Which, finally gave us this graph when plotted alongside our actual observation:
As we can see, our model gave us a slope of -1.884 m/s and we observed a slope of -1.942 m/s, or a 6% difference. Which in the realms of the sorts of things air resistance models are applicible to, is pretty poor (aircraft and such need more accurate measures of air resistance). I suspect that the small ammount of trails we conducted, the relitivly inaccurate way of collecting data, and possibly other factors such as air currents or drafts in the building that we conducted our tests in, have led to this error.
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