Ball+on+incline+Liam+Flaherty+and+Chase+Stockton

__Title of Lab__: Ball on incline

__Researchers__: Liam Flaherty and Chase Stockton

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

__Research__: Variables: Time (t), Displacement (x) Constants: Angle of Inclination (Theta), V i In our experiment, theta = 12 o, and the initial velocity is 0 m/s.

x = V i *t + 1/2*a*t 2 Because V i = 0, this can be simplified to: x = 1/2*a*t 2 Solving for t 2 would yield: t 2 = 2*x/a

However, we want to relate the time to the distance it travels and any constants. Because the acceleration can change depending on the angle of inclination, it would be best if acceleration was not in our equation. The acceleration can be solved for in terms of theta by using trigonometry and the following free-body / component diagrams of the ball: sin(theta) = mg x /mg mg*sin(theta) = mg x Because F N =mg y, the ball is in equilibrium for the vertical forces of the component diagram. To that end, mg x is the only significant force left, and the ball is going to accelerate to the right (the direction of mg x ). Therefore, by Newton's second law of motion: mg x = m*a mg*sin(theta) = m*a g*sin(theta) = a Substituting the acceleration back into the original kinematic equation we previously formulated ( t 2 = 2*x/a) would finally yield: t 2 = 2*x / [ g*sin(theta)] Which is our hypothesis. However, to check our percent error later, we will also solve for g: g=2*x/[t^2*sin(theta)]

__Hypothesis__: t 2 = 2*x / [9.81m/s 2 *sin(theta)] Time squared will form a liner relationship with the distance traveled by the ball. In our case of theta = 12 o, our equation could be rounded to t 2 =.981*x

__Procedure__: Materials: tennis ball, meter stick, stopwatch, incline board, textbook. Note that the textbook must be greater than half the height of the tennis ball, and that the incline board must be wide enough to allow the meter stick and the tennis ball to be on it at the same time.

1. Prop up the incline board such that the longest side of it forms a 10 to 20 degree angle with respect to the horizontal. Make sure that it is not at any sort of an angle where the ball will roll to one side or the other and that it won't move during experimentation. 2. Place meter stick on the side of the incline board to help you determine the distance traveled by the ball. The part of the meter stick closest to the lower end of the incline board should be at 0 cm, with the higher part being 100 cm. 3. Place the textbook at the lower end of the meter stick (at 0 cm). 4. Place the ball on the incline board at a unique location next to the meter stick. Do not let go yet. 5. Record the distance measured by the meter stick at the side of the ball closest to the lower end of the incline board (towards the textbook) in a table. 6. Let go of the ball, and at the same time, start the stopwatch. 7. As soon as the ball hits the textbook (and therefore reaches the end of the meter stick), stop the stopwatch. 8. Record the time in the aforementioned table, next to the distance found in step 5. 9. Make sure the meter stick is still on the side of the incline board and reset the textbook, as it may move when the ball hits it. 10. Repeat steps 4-9 at varying distances until you have a sufficient amount of data.

__Data__: Note that the angle of inclination of our board was 12 o.



__Data Analysis__:

While at first glance the above graph may seem liner, it is still not a good linear approximation of the data, because the average rate of change between each consecutive data point is generally decreasing as the distance gets higher. This is more apparent at either end of the graph. Also, if the line of best fit were to be drawn, the y-intercept would be significantly above the origin. This is incorrect, as the ball shouldn't travel a distance of 0m in about 0.1s if it is moving on an inclined plane. To make this graph more linear, we must square the time elapsed, as shown in the graph below:



This is a much better linear approximation of the data, since there is no noticeably consistent pattern involving the distance traveled and the average rate of change between the data points, and because the data points generally fall into a linear pattern.

The y-intercept (and also x-intercept) is near (0,0), which makes sense since the ball shouldn't move at all in 0 seconds.

The slope of this line is roughly 1.2s 2 /m. We can modify the equation g = 2*x/[t^2*sin(theta)] by using the second graph to become g = 2/[slope*sin(theta)], since it is a linear graph and the tangent line at any point is constant. Plugging in the corresponding values into this equation would yield g = 8m/s^2. Using our experimentally found g and actual g, the percent error is about 18.5%. This is most likely due to the fact that the stopwatch we used could only measure time to a tenth of a second.

__Conclusion__: Due to the level of imprecision of our procedure and the percent error being relatively high, we do not believe our data thoroughly supports our hypothesis. If we were to repeat this lab, two lazer gates (one placed at the start of the ball and the other at the end of the meter stick) that can calculate the time to the hundredths or even thousandths place would significantly improve the precision of our data, and would virtually rule out any extra or lost time between starting and stopping a stopwatch that could measure time to that precision by hand. We would also recommend getting more data, especially at greater lengths than one meter.