Ball+on+Incline-+Sneha+Mittal+&+Michael+Gary

__Title of Lab:__ Ball on Incline

__Researchers:__ Sneha Mittal and Michael Gary

__Research Question:__ How does the time it takes for a tennis ball to roll down an incline relate to the distance that it travels?

__Research:__ Acceleration can be found considering the graphic depicted. Since the ball is rolling on an incline, its acceleration is not due to its weight //mg//, but a component of that weight, //mgx//. Using properties of triangles, one can deduce that **mgx = mg (sin θ).** Newton's Second Law states that the sum of the forces equals mass times acceleration or
 * Σ F = ma **

Assuming negligible friction, one can substitute and simplify to the following equations:
 * mg (sin θ)** **= ma**
 * g (sin θ) = a**

Because one can assume constant acceleration at a constant angle, distance, //d//, and time, //t//, are related in the kinematics equation
 * d = vi t + 1/2 at^2**

In this situation, the initial velocity can be assumed zero, enabling the following simplification:
 * d = 1/2 at^2**
 * t^2** **= 2d/a**

Further simplification allows **t^2** **= 2d/a** to be written as **t^2** **= 2d/(g sin //θ//)**

__Hypothesis:__ The time it takes for a ball to roll down an incline is directly proportional to the distance it travels according to the equation **t^2** **= 2d/(g sin //θ//)**.

__Procedure:__

Materials: plastic board, stop watch, tennis ball, meter stick, protractor

1. A plastic board was placed at the edge of a teacher's cart. 2. A protractor was used to find x, the angle the board made against the cart. Using complementary angles, the angle of inclination was found- 20 °. 3. A distance of 80 cm was measured using the meter stick. 4. The tennis ball was released at the 80 cm distance and the stop watch measured the time it took for the ball to travel the distance. 5. Three trials were conducted. 6. The stop watch was reset and Steps 3 and 4 were repeated using various distances.

__Data:__

The three trials produced the following results:
 * || Time (s) Trial 1 || Time (s) Trial 2 || Time (s) Trial 3 ||
 * Distance (cm) ||  ||   ||   ||
 * 80 || 0.78 || 0.72 || 0.71 ||
 * 70 || 0.67 || 0.65 || 0.66 ||
 * 60 || 0.62 || 0.62 || 0.63 ||
 * 30 || 0.41 || 0.43 || 0.36 ||
 * 20 || 0.36 || 0.40 || 0.41 ||

The following values will be used in the graph, based on precision: __Data Analysis:__
 * Distance (cm) || Time (s) ||
 * 80 || 0.7 ||
 * 70 || 0.65 ||
 * 60 || 0.62 ||
 * 30 || 0.4 ||
 * 20 || 0.4 ||

The graph of time plotted against distance (experimental values) takes the following shape:

When time squared is plotted against distance, the best fit line is linear.

The y-intercept in both graphs can be assumed zero since it takes zero seconds for one to travel zero distance. Using the linear regression formula given by Microsoft Excel, the average rate of change of the Time^2 vs. Distance is 0.0062 s^2/cm or 0.62 s^2/m. Using the equation, g= 2/(slope*sin θ), the experimental g is calculated to be 9.4 m/s^2; thus the percent accuracy is determined to be 96%

__Conclusion:__ Despite the percent accuracy, one can not conclude that the experimental results support the hypothesized equation. The data is not sufficient to draw a proper conclusion, lacking test points between 30 cm and 60 cm due to inadequate time; thus, the experiment is left inconclusive. In future experiments, the procedure could be revised to incorporate a greater number of test points at consistent intervals so as to determine a more accurate trend.

Proper experimentation and data analyses lead to a greater understanding of the physical forces governing the universe. Understanding these forces enables enhancement in machinery and efficiency, leading to improved lifestyles and greater prosperity.