Golfball_6

=**__Golf Ball Dropped From 6th Floor of Atrium__**=


 * Time (sec) || Distance (cm) ||
 * 0.16 || 12.1 ||
 * 0.24 || 29.8 ||
 * 0.25 || 32.7 ||
 * 0.30 || 42.8 ||
 * 0.30 || 44.2 ||
 * 0.32 || 55.8 ||
 * 0.36 || 63.5 ||
 * 0.36 || 65.1 ||
 * 0.50 || 124.6 ||
 * 0.50 || 129.7 ||
 * 0.57 || 150.2 ||
 * 0.61 || 182.2 ||
 * 0.61 || 189.4 ||
 * 0.68 || 220.4 ||
 * 0.72 || 254 ||
 * 0.72 || 261 ||
 * 0.83 || 334.6 ||
 * 0.88 || 375.5 ||
 * 0.89 || 399.1 ||

By plotting the data into the calculator, with time in the x and distance in the y, the resulting scatter plot would look like this: We figured out that this piece of data is a power function. To find a power model, we first calculated it by hand.

First, pick two points that fits the graph the most. (0.72, 254) (0.36, 63.5)

Substitute the two points into the parent power function of:



By using PWRReg in the calculator we get:

By plotting the equation found by hand, you can see that the data fits pretty well with the power equation. By looking at the residual plot, it is clear that the fit for this function is faily good. All the points are close to the x axis, and all the points don't show a certain pattern.

By plotting the equation found by using the PWRREG button on the calculator, you can see that the two different equations found are very similar to each other.

However, by using PwrReg in the calculator, it give the equation of y= 491.42x^1.9867, with a correlation coefficient of 0.998, showing that the equation fits the data 99.8%.

By looking at the two graphs, it's hard to tell which one is the better of the two. However, by looking at the residual plot, it gives us a better idea of how well these graphs fit the data. After close examination of the residual plot, it is clear that the equation found by hand is more accurate of a presentation of the data than the equation found by the calculator. Plus, both residual plot graphs don't show a pattern, thus making it an even better fit. As you can see for yourself, more of the points in the hand found residual plot graph is on the x axis than the other residul plot graph. Thus meaning, the points are closer to the line. Besides, it makes sense that this graph is more accurate due to the numbers. 491.4 seems like a more accurate number than a simple 490, and 1.987 although rounds up to 2, obviously makes a slight difference that makes this graph the better of the two.

By calculating the residual sums of the two graphs, we come up with Residual sum of Residual sum of

Thus, we can see that the equation found by the calculator is a better fit.

__**Comparison of Functions**__


The two functions, one obtained by hand, the other through the PwrReg utility in the calculator are both good models for the data. As you can see, the two equations share a very small difference. This can be attributed to the premature rounding done when finding the equation by hand. A lot of steps are not as convenient for us to do on pen and paper when compared with a calculator fitted with storage capacities and a whole array of other tools. Still, we cannot tell whether this minute difference between the two graphs make either one better without checking the residual plots. Upon further investigation with the Residual Sums of both equations, we see that the calculator’s fit is in fact closer and more accurate than our equation found by hand. Residual sum of Residual sum of

__**Extrapolation:**__ Using the above model, we are going to use a few points outside of the data region to predict what might happen in the future.

 * x = [1, 1.5, 2, 10]**









Assuming all floors of the Atrium are the same height, and that each floor is 3.5 meters long (average standard height), 6 floors is 21 meters. Which is 21000 centimeters.
 * y = [21000]**

Assuming the above two conditions is true, 6.62 seconds would be an appropriate guess for how long the golf ball takes to reach the bottom floor. We can easily modify the y-value to find out how long it would take the golf ball to reach the bottom but because we have only been offered with one piece of information about the floors and the total height (6 floors) we must make-do with our assumptions and predictions.

Let us define t(k) = some time known in the series, then f(t(k)) is the output at time t(k) and the doubling time would be T(d) = 2f(t(k)). and let 2f(t(k)) = f(t(delta)) such that t(delta) > 0 ⊆ All real numbers.
 * Doubling Time:**

The following is to solve for t(delta) or in other words the rate of doubling time.

With this equation, you can find the doubling time for any known y-value (distance). When this equation is plotted, a linear graph appears, and it shows that the doubling time, as the y-value increases, decreases and is achieved proportionately faster than previously.

Linearizing the Data:

As you can see from the graph above, by logging the x and the y, we straightened out the data. However, we tried others just to make sure that this was the best one.

Using LinReg from the calculator, it gives the equation of y= 1.987x+ 2.691 for the linearlized data. With the correlation coefficient of 0.998, it is said that the equation fits the data 99.8% which is very close.

By logging y and leaving x as it is, we see a curved line, thus meaning that the data was not exponential. By logging x and leaving y, we still see a curved line, thus again meaning that this data was not exponential.

As a result, we conclude that we must log both the x and y axis to linearize our data, therefore proving that this data exhibits a power function. This data could be represented by the equation derived from the calculator: y= 491.42x^1.9867. This equation is a good fit as the correlation coefficient is 0.998, resembling a 99.8% fit of the equation to the data. Our linearlized equation derived from the calculator is :y= 1.987x+ 2.691. This equation is also a very good fit for the linearlized data as the correlation coefficient is of 99.8%.



Both semi-log graphs are curved models and technically the Log-Log Graph is right in the middle of the two of them. Unlike other data, this one balances out perfectly between the three graphs. Once again though, the Log-Log graph is the one that linearizes the data, proving that this is indeed a power function.

Related Topics: [|Tutorial on Power Functions]