Freefall

=__Free Fall Investigation --Chloe Lin__= In this investigation, our goal is to find a function that models the height of an object falling due to the force of gravity.

By doing this, we used a cushion as our falling object, and used a motion detector to obtain the data. Raising the cushion 2 meters above the motion detector placed on the floor, we dropped the cushion directly above the detector and came up with a good set of data. Since the motion detector picked up everything from the stationary moment right before we dropped the cushion, plus the bounce that occured after the fall, we needed to pick out the data that we needed to proceed in our investigation.

Although the motion detector picked up a lot of information, we didn't need the whole part, thus we chose the appropriate data points to represent the fall.

This is what the data looked like after we picked out what we needed, as you can see it has a nice curve to it.


 * x (time in seconds) || y ( height in m) ||
 * 0.915 || 1.940 ||
 * 0.968 || 1.939 ||
 * 1.021 || 1.916 ||
 * 1.075 || 1.869 ||
 * 1.129 || 1.796 ||
 * 1.183 || 1.699 ||
 * 1.237 || 1.576 ||
 * 1.290 || 1.437 ||
 * 1.344 || 1.281 ||
 * 1.398 || 1.098 ||

Using the Finite Difference Method, this is what would happen:

When I got to about the second difference, I saw a reoccuring number that revolved around -0.024, thus I concluded that a 2 degree function would be a good function for this data. We can check this by using the finite difference method on the calculator as well. When we did it the first time, the trend of the data graphed was a nonhorizontal pattern. But after two times, the calculator came up with a graph that showed the set of data had a horizontal linear trend, it emphasized the fact that the data we have can be modeled with a 2degree polynomial.

To find this equation that models out data, we needed to do a series of steps. According to the Reference Chart, this is what would happen: 2a= -0.024 9a+b= -0.097 25a+5b+c = 1.699

With that, we result with: a= -0.012 b= 0.011 c = 1.944

Thus, we get an equation of y = -0.012x^2+0.011x+1.944 Yet, when we checked this on the calculator, it was actually 2 very different equations. By using QuadReg. on the calculator. We get : y = -3.931x^2 + 7.335x - 1.478

It seems that the equation we found by hand did not match up with the equation on the calculator. The calculator is obviously the better fit. The line we obtained from the finite difference seems to be a constant line. On the other hand, the calculator goes through the data points nicely. Yet, based on these results, it is apparent that the degree of the polynomial function that models free fall is two nonetheless. With that, I would say the general form of this polynomial function is y = -3.931x^2 + 7.335x - 1.478 seeing that it fit the data the best.

Using what we learned on projectile motion, we can verify our model.

y = -0.5g t^2 + V0 sin(beta) + Y0 y = -0.5gt^2+Y0 Since sin(90) = 1, and we dropped it vertically. Since we need to know how long the cushion took to hit the ground, we took the last x value and subtracted the first x value, resulting with 0.483 seconds y= -0.5(9.8)(0.483)+Y0 Making y = 0, we can see check our data. 0= -0.5(9.8)(0.483)+Y0 0= -4.9(0.483) + Y0 2.366 = Y0

Our Initial height (Y0) was suppose to be 2 meters, although 2.366 is a little off, we could say that it was close. This little difference could be caused by the fact that the calculator did not exactly collect data in arithmetic intervals, thus resulting in this slight difference. Yet, since our data did some what fit the projectile motion equation, we could say that our model does predict an object free falling.

Conclusion: By using finite difference method, we can clearly see how the polynomials of a function is obtained. For this investigation, we ended up with a 2 degree polynomial. Moreover, by comparing it with what we learned from projectile motions, we could compare the results with our free fall investigation. In the end, this showed us that the data we collected did reflect a free falling motion, and the function we got in the end did model it.