Free Fall lab report, According to the date provided and instruction attached

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Free Fall

Introduction

This experiment is designed to study the motion of an object that is accelerated by the force of gravity. It also serves as an introduction to the data analysis capabilities of Capstone, and the calculational and graphing capabilities of Excel. To observe the relationship between the position, velocity and acceleration of a falling object, you will record the position of a bouncing ball over time and calculate the resulting velocity using Capstone and Excel.

Reference

Young and Freedman, University Physics, 12th Edition: Chapter 2, section 2.5

Theory

A free falling object is accelerated toward the earth with a constant acceleration equivalent to g, 9.8 m/s2. The velocity, v, in meters per second is therefore changing according to its acceleration. We wish to test the kinematic equations for a free falling object:

1.1

1.2

The vertical position (height) is given by y, with respect to time, t, where y0 is the initial vertical position, v0 is the initial velocity and g is the acceleration due to gravity. The plus sign in front of the last term in the equation is due to the coordinate system used in this experiment. This equation yields a quadratic curve for the position vs. time. The relationship between velocity and time can be plotted as a straight line with a slope, g and a y-intercept of v0.

We will also need the definition of average velocity:

1.3

In the case of a bouncing ball, acceleration is constant while the velocity and position change over time. So for each bounce, the slope of the velocity versus time graph should yield a relatively constant value of g. To evaluate how constant these values are, you will measure 10 slopes and calculate the standard deviation. The standard deviation is a measure of how much the data deviate from the average value of the set.

Procedure

In the experiment you will use the Capstone motion sensor to record at least 10 bounces of a ball that is dropped from some height near the motion sensor. The motion sensor is mounted on the edge of the table facing down.

Part I: Determine g.

1. Check to make sure that the Motion Sensor is set to the correct distance range for today’s lab. To do this look at the top of the sensor and make sure it is on the short-range designation. See Figure 1.1, where short-range is on the right. If you have trouble later, try switching it to long-range.

Figure 1.1

2. Plug the yellow and black wires into the (1) and (2) digital channels, respectively, on the 850 Interface Box.

3. Double click on the Pasco Capstone icon on your computer desktop. Double Click Graph under the Display menu on the right hand side.

Figure 1.2

4. Go to the tool bar at the top of the page. Click the Add New Plot Area tool once to create two graph areas, for position and velocity as functions of time.

Figure 1.3

Figure 1.4

5. Click on Hardware Setup icon in the left vertical toolbar, and then click on the left circle in the upper image (that correspond to the yellow cable input on your Capstone box), and select Motion Sensor II.

6. Label the Y axis of the top, middle, and bottom plot area to position and velocity, respectively

7. In the lower box, next to recording conditions set the sample rate of the Motion Sensor II to 40 Hz.

8. Hold the ball about 15 cm below the motion sensor. Click on the red Record button on the bottom of the Capstone screen. When the timer begins to show the passage of time, then release the ball.

9. After the ball has bounced as many times as possible, press the Stop button in PASCO Capstone. In the end, you will want a total of 10 bounces but these may be taken from different trials as you will most likely not get 10 usable bounces in one trial.

10. Since the motion sensor is located above the ball, the sensor gives an inverted coordinate system. The position 0 meters is always located at the sensor and the sensor records distances as more and more positive as objects are moved farther and farther away. (This also means that velocities will be negative as objects move toward the sensor and positive as they move away.) You can see from figure 1.3 that the ball drops from its highest point at about 0.15 m from the sensor, bounces, landing on the floor at 0.9 meters from the sensor and returns to a height of 0.5 m from the sensor before dropping back to the floor at 0.9 m and so on.

11. The equation that relates position and time for motion in one dimension is equation 1.1 given in the theory section earlier. From it, we would expect to see quadratic shape for the curve for each bounce as shown in the upper half of figure 1.5.

Figure 1.5

12. For the velocity vs. time graph, note that for each section where the ball is in the air (one “bounce”), this graph is a straight line as predicted by equation 1.2. It should be noted that the acceleration is positive because the location of the motion sensor gives an inverted coordinate system as explained in step 9. (Gravity is accelerating the ball away from the motion sensor which gives positive position and velocity values, as well.)

13. The starting point of each straight section is the rebound velocity after the bounce, and the end of the straight section is the velocity with which it collides with the floor at the next bounce. Click the Highlight Data icon to create a rectangle that will select data points. Move the rectangle and adjust it around the section of your velocity plot where the ball is in the air. The selected data will be shown highlighted.

14. Click the Fit icon on the toolbar above the graph. Select Linear Fit from the drop-down menu to display the slope of the selected region of your velocity vs. time plot. The slope of this part of the velocity vs. time plot is the acceleration due to gravity during the selected region of motion.

Figure 1.6

15. Find the slope of the velocity curve for each bounce by selecting the appropriate range of data and following the procedure above.

16. Make a table in Excel that shows these slopes.

17. Find the average value of g for the 10 bounces you recorded in the table in Excel. To do this, select a cell and type ‘=AVERAGE(A1:A10)’ assuming that your g values are in cells A1 through A10. If not, simply type in the appropriate range to calculate the average.

18. Calculate the standard deviation for g (σg) by selecting a cell and typing ‘=STDEV(A1:A10)’. Again, if your data are in a different column, simply adjust the range of your data.

19. Now calculate the standard error of your average g value using equation 1.9 of your Measurement Uncertainties handout. Does your average value for g agree with the accepted value within the standard error (σave)?

Part II: Error analysis

To help you understand what the Capstone software is doing to calculate the slopes of the lines that you used for your g values in Part I, you will export the data for one bounce into Excel and perform the analysis yourself.

20. Export the position vs. time data for one bounce into Excel by highlighting the data of interest in the position versus time graph. Double click the table icon underneath the display toolbar on the right hand side. Select the first column as “time” and the second as “position”. The highlighted data on the graph should already be highlighted in the table. Copy (Ctrl C) the data from the table and then paste (Ctrl V) it into Excel. The time data should appear in column “A” and position data should appear in column “B” with the first data point in the same row as the time data.

21. Plot the position vs. time in a scatter-plot. Label your axes with proper units.

22. To calculate the velocity, you need to skip the first row of the column corresponding to the velocity. Enter the equation for velocity (equation 1.3). Once you enter this equation, you will see the velocity for the second point. To obtain velocities for subsequent points, click on the lower right-hand corner of the C3 cell, hold it, and drag the cursor down the column to the last row of your data. Do you understand why you will have one fewer data point for velocity than position? This should copy the formula to all of the cells in the column that you highlight. Excel will automatically advance the numbers in the formula down the column.

23. Plot the velocity versus time in a scatter-plot with a linear trend-line. Display the parameters of the trend-line and compare these results with the Capstone values for the slope of the velocity versus time graph. Determine the values of v0 and g using the trend-line. Does your g value agree with the value that Capstone gave for the same bounce?

24. Before you print out your graphs, right click on the graph and select Format Plot Area. Select none (or no Fill) under Area and hit OK. This will save ink by giving the graph a white background and should be done for all graphs plotted in Excel throughout the semester.

25. To measure the accuracy of the motion sensor, point the motion detector at the ball sitting still on the floor and record the position for a few seconds. Put the position data into a table by dragging a table icon to Position. Click on the top of the table to select all of the data and then copy and paste it into Excel. When the data is in Excel, use the standard deviation function to calculate the standard deviation of the position. The standard deviation of the position gives the uncertainty of position in the y direction (σy).

26. Using equation 1.2, calculate the uncertainty for your value of g. Find your initial and final velocity from your Excel table. Find the uncertainty in each by propagating uncertainty using equation 1.3. Solve equation 1.2 for g and propagate uncertainty to find σg. Assume 1% uncertainty for your time values. This is a multi-step calculation that should be shown in detail in your lab report.

For your lab report

Make a result table and show the results found in steps 18, 19, 23, 25, and 26. Include your graphs from steps 20 and 22.

In your Calculations section, show a sample calculation for the velocity calculation in your Excel table (Step 22). (You should always show a sample calculation for any calculated column in an Excel file in all labs this semester.) Show the calculations from step 26.

In the discussion part of your report discuss the shape of each graph that you made in this lab and what information it contained. Did your average value for g found in step 17 agree with the accepted value within one standard deviation? (Use the standard error from step 19 for this comparison.) How does the standard deviation for g found in step 18 compare with the uncertainty estimate from step 26? Does your uncertainty estimate give you a good idea of the actual variation of the g values?