Marble Launch Experiment Purpose The intent of this experiment is to numerically trace the path of a marble launched into the air at an angle in order to observe the parabolic nature of the trajectory. After data acquisition, several features can be observed about the trajectory such as; the constant acceleration of gravity, the nearly constant horizontal component of the marble s velocity, and the effect of air resistance. Procedure In order to accurately see the instantaneous position of the marble, a video camera will be used to capture frame by frame pictures of the marble as it travels along its course. First the camera must be positioned in a stand to hold it stationary, so that movements of the camera do not affect measurements taken. Make sure to use some sort of single colored (preferably white) background so that the marble stands out in the video, also some sort of scale or ruler should be included in the background so that measurements can be made in standard units. Some sort of mechanism should be used to launch the marble in a consistent manner (as opposed to throwing it) so that the plane of the marble s path is perpendicular to the camera s line of sight (this minimizes effects of parallax). This should be tried several times to ensure that the marble stays in the frame also. When the launcher and camera have been adjusted to properly account for all of this, record the marbles trajectory. Once the shot has been recorded digitally, import the video into Logger Pro. This software has the capability to extract usable data from digitally recorded video (with some manual intervention, anyway). In order to do this you must establish and orient the origin and coordinate axes so that the marble s position can be referenced to it. Also set the software s scale to match that of the background. Once this is done, plot the position of the marble in each frame of the trajectory. The software will keep track of the positions and times at which they occurred in the video and can calculate quantities such as velocity and acceleration. You can also export the data for use in other software for graphing/analysis.
Results Position (cm) Y Position Over Time 50 45 40 35 30 25 20 15 10 5 0 9.7 9.8 9.9 10 10.1 10.2 10.3 10.4 Although the graph is not a true parabola due to effects of air resistance, it comes very close. Since the marble has a small surface area to mass ratio compared to many other objects, air resistance is nearly negligible. From this data, the acceleration of gravity can be approximated using a quadratic regression. Using SigmaPlot statistical analysis software to perform the regression, I get a formula that roughly represents the whole trajectory: y = 49932.408 + 9929.464 t 493.175 t 2 Without air resistance, the vertical component of a projectile follows the time dependent equation: y = y 0 + v y0 t 1 2 gt2 In order to obtain the acceleration of gravity from this, we solve for g in the general equation from the regression: 493.175 = 1 2 g g = 986.35 cm Or g = 9.8635 m This is very well within the range of acceptability, since the generally accepted value for g is about 9.81 m/, and this value varies depending on the latitude and altitude from which it is measured (it also has
a slight time dependence due to varying mass distributions within the earth). If we take the actual value to be 9.81m/, then we can find the percent error in the measurement: %error = g actual g measured g actual 100% = 9.81 m 9.8635 m 9.81 m 100% = 0.545 % The calculated value of g could be very much improved upon if the complete apex of the trajectory was captured by the camera. At the apex of the flight, the force due to air resistance is at a minimum, since the marble s velocity is at a minimum (and air resistance increases with velocity). If the regression was performed on just the points at the apex of the flight, then the calculated gravitational acceleration would be more accurate since gravity would be more dominant over air resistance at those points. Another way to arrive at the value for g is to use the y component velocity of the marble as it flies through the air. Since gravitational acceleration is constant, the value for the y component of its velocity is a linear function of time. In order to find the Y velocity of the marble component at a particular moment, we divide the difference between two consecutive y positions by the time interval between them. This results in the average velocity in that time interval: v Y,avg = Δy Δt Doing this for every pair of adjacent Y values, the following graph is obtained:
300 Average Y Velocity Over Time Velocity (m/s) 200 100 0 9.7-100 9.8 9.9 10 10.1 10.2 10.3 10.4-200 -300-400 Preforming a linear regression on this data set results in the equation: v y = 9850.763 981.114 t In general, the y component velocity of a projectile without air resistance is: This indicates: v y = v 0,y gt g = 981.114 cm Or g = 9.81114 m This value is so close to the generally accepted average value of 9.81 m that calculating the percent error is meaningless, since the accepted value is not precise enough to gauge inaccuracy against. Yet another way to arrive at g, not using statistical regression, is to calculate the average acceleration between points on the velocity graph. Using essentially the same process that the average velocity was calculated by, average acceleration can be calculated using the formula: a y,avg = Δv y Δt Repeating this process for every point on the velocity graph yields the average acceleration graph:
Average Y Acceleration 0 9.7 9.8 9.9 10 10.1 10.2 10.3 10.4-500 Acceleration (m/ ) -1000-1500 -2000-2500 In ideal conditions, this graph should be a horizontal line. Due to factors such as measurement error and unaccounted air resistance, it is far from flat. The fact that air resistance is dependent on velocity causes deviations from the constant g. Taking the average of all of the data points gives some sort of idea of the magnitude of constant acceleration that occurred throughout the marble flight. This average works out to be a = 973.598 cm Or a = 9.73598 m This value has an error of 0.754% compared to the accepted value of 9.81m. While this result is the further from the accepted value of g than the other two methods, it is still within the range of acceptability. The fact that three different methods of calculation of g yield a value that is within 1% of the accepted value is a good indication that the measurement process was fairly accurate. Without air resistance, the x component of a projectile s velocity should be constant. This can be seen in the (very nearly) straight line graph of the marble s X position:
X Position Over Time 40 30 Position (cm) 20 10 0-10 9.7 9.8 9.9 10 10.1 10.2 10.3 10.4-20 -30 The very slight concavity of the line is most likely due to air resistance. In a the same manner that the Y velocity component was obtained, the X component can be obtained just as easily: v X,avg = Δx Δt After performing this operation for every pair of adjacent X values, the following graph is obtained: 115 110 Average X Velocity Over Time Velocity (m/s) 105 100 95 90 85 9.7 9.8 9.9 10 10.1 10.2 10.3 10.4 In an idealized model, this graph should be a horizontal line, but it is obviously not. In fact, the graph is very convex. The shape of the trend is characteristic of air resistance. When air resistance is directly proportional to velocity, the following equation holds true for an object experiencing drag:
v = v 0 e bt m Where b is a constant of proportionality between velocity and force of air resistance, and m is the mass of the object, since it is derived from the differential equation: m dv dt = bv When graphed with arbitrary values of v 0, b, and m, the former equation has the shape: Using an image editing program to overlay the graph of the data and a section of the air resistance curve (while still preserving aspect ratio) yields the following diagram: V el o ci t Time (S) When the two are placed in the same frame, it becomes obvious that the x velocity data obtained from the video appears to follow this trend line, deviating greatly from the expected horizontal line. This
indicates that even though the marble has a small surface area to mass ratio and experiences relatively little air resistance, the measurement process was sensitive enough to detect its effect. It would be interesting to attempt to manually fit the curve to the data (as opposed to just superimposing the two) in order to approximate the value of b given a measured value of m, the marble s mass. Although this would be inherently inaccurate, since the drag experienced by the marble might not exactly be proportional to velocity, it should be in a ballpark range. Air Track Experiment Purpose The purpose of this experiment is to record a slider on a frictionless air track with a video camera in order to extract position data from the recording. From this data, apparent velocity can be deduced. Procedure In order to get data for this experiment, a digital video camcorder is needed. This is so that the video recording from the camera can be directly imported into software that can turn the video into useful data points. Set up the video camera so that it points directly perpendicular to the lab table and the air track. Keeping the camera as perpendicular as possible ensures that parallax errors in the video are minimized, which affect apparent position and determination of velocity and acceleration. Push the slider at a reasonable velocity at which the slider will neither appear as a blur in the video (because it is moving too fast) nor take an excessive number of frames to move a small distance (because it is moving too slow). When a reasonable velocity is found, press the record button on the camera to capture the slider as it moves in and out of the camera s view. Import the video into Logger Pro, set the scale, set the origin, and collect position data points. Results
Position (cm) X Position Over Time 50 45 40 35 30 25 20 15 10 5 0 38 38.5 39 39.5 40 40.5 41 The graph of the x position of the slider is a straight line. This is indicative of a constant velocity. The slope of this line would be the velocity of the slider. Preforming a linear regression on the x positiontime data yields the equation: x = 704.14 + 18.42 t Under ideal conditions with no acceleration, the position of an object as a function of time is given by the formula: x = x 0 + v t This means that the velocity of the slider is 18.42cm s or.1842m s. Another means of arriving at the velocity of the slider is to take the average velocity between position points by the formula: v avg = Δx Δt After performing this operation for every pair of adjacent X values, the following graph is obtained:
Velocity (m/s) X Average Velocity 21 20.5 20 19.5 19 18.5 18 17.5 17 16.5 16 38 38.5 39 39.5 40 40.5 41 With no errors in measurement and assuming that there was no external acceleration in the experiment, the graph should be a straight, horizontal line, since the velocity of the slider should be constant. However, the graph makes several unexpected jumps. Even with these jumps, the graph still seems to follow a horizontal trend line in the beginning of data collection. Toward the end of the data set, the velocity seems to decrease. This is most likely due to a slight parallax error that is caused by the viewing angle of the camera. Since the velocity data is not as contiguous as that obtained for the marble launch, effects such as parallax and air resistance are harder to recognize. Notice also that the jumps in the graph seem to be quantized. That is, they seem to end up in recurring y values. This could be due to the data point acquisition process, in which the position of the slider is manually selected by placing the mouse on pixels in the video. Since the video does not have an infinite resolution, a single point on the slider will jump forward and backward compared to its expected position. Simply put, a point on the slider cannot move less than a pixel. The same principle applies when a point on the slider is selected; you cannot actually plot exactly the same point on the slider because that point could be in between pixels. Also, there is a significant human error in actually selecting the points; it is hard for one to tack a single point for many frames with exact accuracy. As another method of calculating the velocity of the slider as it passed through the frame, the average of the points on the X Average Velocity graph can be taken. This calculation yields the value: v = 18.39 cm s Or v = 0.1839 m s The percent difference of the two values obtained for the measurement of velocity is:
% diff = v 1 v 2 v 1 + v = 0.192% 2 2 The fact that difference between the two measurements of velocity is less than a fifth of a percent seems to indicate that the values obtained for v are fairly accurate. This experiment could have been more accurate if the camera was positioned at a better viewing angle. It is evident in the video that the camera was not exactly perpendicular to the track. Also since the camera was on a rigid stand, vibrations from the table and the act of pressing the record button remain present for a long time in the video. If these problems were ameliorated, then more useful data could be extracted, allowing for a more precise calculation of v and for the determination of other effects such as parallax and air resistance.