Sonic Ranger

3.6.1 Dependence on Mass

The effect of mass on acceleration due to gravity was tested using balls of identical size and shape. One sheet of paper was crumpled into a paper ball. A second sheet was then cut in half and one of those halves was crumpled into a second ball. Finally, the second half was again cut in half, making quarters. One of the quarters was crumpled, making a total of three balls made from paper. The balls were condensed to a size as close to one another as possible. For a fourth ball, a tennis ball was used. The full size sheet weighed 5g, the half sheet weighed 2.5g, the quarter sheet weighed 1.25g, and then tennis ball weighed 55g. The apparatus that allowed this experiment to happen was the sonic ranger, place on top of a 2-meter hollow tube. The ranger was connected to a computer program, MiLab, which generates velocity and acceleration data from the sonic ranger's distance vs. time data. Because the data looked suspicious at first, a foam pad was place underneath the tube. The paper balls were dropped a total of 5 times each, while making sure they did not touch the side of the tube. The tennis ball was dropped only once. The mean acceleration, standard deviation of the acceleration, standard deviation of the mean of the acceleration, and the slope of the velocity vs. time graph were taken for each trial. The following page is the data chart that contains all of the data taken, and the page after that is the mean acceleration vs. mass graph.

3.6.1 Analysis

The data chart on the following page summarizes all of the data from 3.6.1. In order to see the trends in data properly, the mean acceleration vs. time graph should be referred to on page 3. The maximum standard deviation was 4.5 m/s^2. This is the standard of deviation used for all points of the graph, even though three points have lower standard deviations. There is a general trend between the data - as the mass increases, so does the acceleration. The graph on page 3 illustrates this. The tennis ball, with much more mass than any of the paper balls, has a much higher acceleration. Looking at the graph, one could infer that no matter how large the object, freefall acceleration will be no greater than 9.8m/s^2, the force of gravity.

There are two forces acting on the objects falling. The force of gravity and the force due to air resistance are these two forces. All objects in freefall will have the same acceleration. However, the weight of the object is counteracted by air resistance. Objects that have the same size and shape have similar air resistance. However, objects of a lighter mass, however, have less weight, and thus are affected by air resistance more than objects with greater mass. The result is a slower acceleration.

Possible errors in this experiment could be the time delay from when the masses were dropped. Also, as the mass exited the tube, the sonic ranger's ability to generate accurate data may be compromised. The size of the paper balls were generally the same, but it was impossible to make the balls exactly the same. Also, the actual surface of each ball of paper probably differed a lot, so the effect of air also differed. The sonic ranger may not have been calibrated properly, although this is unlikely. This is considered a systematic error. Also, the objects may have hit the side of the tube, which would change the acceleration because of friction.

3.6.2 Bouncing Balls

An attempt was made to bounce a tennis ball on the surface of the floor, in hopes of attaining a graph that would show the path a ball takes when it bounces. Unfortunately, it was too difficult to get the ball to bounce back up into the tube, so the foam pad was kept under the tube. The tennis ball was then dropped and the sonic ranger showed the path the ball took. From this graph, the acceleration a) after dropping the ball, b) when the ball hit the ground, c) after the ball bounced up, d) when the ball reached the apex of its bounce, and e) after the ball reached the apex of its bounce were all calculated.

Calculations

A) (1.9-.44)/(.9-.5125) = 3.77m/s. (3.77m/s/(.9-.5125) = 9.72m/s^2.

B) (1.85-2.00)/(.925-.88) = -.012m/s. (-.012m/s/(.925-.88) = -.398m/s^2.

C) (1.9-1.65)/(1.1-.9) = 1.25m/s. (1.25m/s/(1.1-.9) = 6.25m/s^2.

D) (1.65-1.625)/(1.1-1) = 0.30m/s. (0.30m/s/(1.1-1) = 3.00m/s^2.

E) (1.875-1.65)/(1.325-1.1) = 1.00m/s. (1m/s/(1.325-1.1) = 4.44m/s^2.

*Graph on the following page, all data in chart on page 2.

3.6.2 Analysis

As expected, the acceleration during the initial drop was near the acceleration due to gravity. The mass of the tennis ball was enough to overcome much of the air resistance on the ball. As the ball is in contact with the ground, a very negative acceleration is expected. However, because the foam pad was kept underneath the tube, the deceleration of the tennis ball was much less than it would be if it had hit the ground. The acceleration on the way up after the first bounce was 6.25m/s^2. This is probably low because the foam pad absorbed much of the force that the ball would have kept had it hit the ground directly. Therefore, the ball had a lower acceleration. At the top of the bounce the acceleration was 3m/s^2. Finally the velocity while the ball was falling from the apex of its bounce was about 4.44m/s^2.

The fact that the pad was kept underneath the tube may have skewed the data a little because the acceleration after the bounce should probably have been a little bit higher than the acceleration of gravity, because the while the ball is moving upwards air resistance is down, which is the same direction as gravity. Hence, the acceleration should be greater than 9.8m/s^2. However, the foam pad kept it from accelerating back upwards. Unfortunately, because of time restraint, persistence in trying to properly bounce the tennis ball back up into the tube after the bounce was not an option. Had there been more time, and the ball could have been bounced back up the tube, then the acceleration would have been much higher throughout the rest of the path the ball took.

One possible source of error, although unlikely, is that the sonic ranger wasn't properly calibrated. The most problematic part of this experiment was trying to drop the ball such that it 1) did not hit the side of the tube and get affected by a frictional force and 2) bounced straight back up into the tube again without hitting any of the sides. Also, as the ball