Nonuniform Motion Lab
Purpose
The purpose of this lab exercise is to investigate the motion of objects that do not have a constant acceleration and to establish the validity of velocity as a derivative of position and acceleration as a derivative of velocity.
Procedure
In this experiment you will measure an object moving vertically above a Calculator Based Ranger (CBR). As the object moves up and/or down the CBR will collect data about the object’s position, velocity, and acceleration. The data will be fed via Universal Lab Interface (ULI) into a laptop computer running Logger Pro software. This program will be used to print graphs of position, velocity, and acceleration, and to determine the best fit equations thereof. The goal is to fit appropriate polynomial equations to the same time interval of each graph. Based on your knowledge of derivatives it should be evident that the polynomials of the graphs will be of different degrees. For example if the position graph is a degree 4 polynomial, then the velocity graph will be a degree 3 polynomial and the acceleration graph will be a degree 2 polynomial.
Part A – Motion of a Hand!
1. Make all connections BEFORE turning the equipment on. The CBR is connected to Port 2 on the front of the ULI. The ULI is connected to the serial port on the back of the computer. Once all connections are secure, and only then, turn on the ULI and turn on the computer. At the log on screen enter name: student, password: student.
2. Start the Logger Pro program and open the file called “Nonuniform”.
3. The file contains formatting information that sets up the CBR to measure position, velocity, and acceleration. Once the file is opened, the CBR is ready to collect data. Simply click on the Collect button. If the results are not satisfactory just run it again – the old data is automatically deleted and new data is collected each time you click on the Collect button.
4. To perform the experiment your hand should move up and down in a line above the CBR. Click on the Collect button, wait until you hear the CBR operating, then move your hand up and down smoothly. It is best to have your hand palm down with fingers straight out to form a flat surface from which sound waves can reflect.
5. If the experiment is successful you should now see three graphs that clearly illustrate the position, velocity, and acceleration of the hand. Select a graph and use the autoscale tool button if the data goes off the edge of the window. If there are significant glitches in any of the three graphs then simply collect new data. All three graphs should be smooth, continuous curves. Note: the CBR cannot measure positions less than about 40 cm (16 in).
6. Once you have an acceptable set of data you are ready to analyze the graphs.
7. It helps to start with the acceleration graph. Note: you can maximize its window for a better view. Find a section of this graph consisting of at least 10 data points that can be fit to a degree 2 or 3 polynomial equation. To do the curve fit, use the mouse to select the section of the graph in question, under the Analyze menu choose Automatic Curve Fit, and then choose the appropriate equation type and click on Try Fit. If you are not happy with the fit (it should be a close match to the data) then remove it and try a different selection of data on the graph or, if necessary, collect new data.
8. Repeat the steps outlined in the previous step to find appropriate polynomial curve fits on the other graphs for the same interval of time analyzed on the acceleration graph. To find the same interval of time note that the coordinates of the mouse pointer can be found in the lower right-hand corner. Also you can use the x = ? “examine” tool to trace along the data points.
9. Once you have curve fits for every graph you are about ready to print. Under the File menu choose Printing Options and enter name and comments. Under the File menu choose Page Setup and check for appropriate choices. Click on the graph you want. Maximize its window. Under the File menu choose Print Window (not Print Screen). Repeat for each graph.
Part B – Motion of a Balloon!
1. Clear the data from part A. Under the Data menu choose Clear All Data.
2. Repeat the process used in Part A with a few modifications:
3. Obtain a balloon and blow it up to about the size of a human head. Tie it off.
4. Throw the balloon gently upward above the CBR and catch it as it returns. It is best to have your hands on either side of the balloon and to move them out to the side while the balloon is in the air. This is so that the CBR does not measure your hand’s motion instead of the balloon’s motion.
5. Find the interval of time where the balloon is airborne and scrutinize the acceleration. It should be possible to find a fairly linear section of data in this region. Try a linear fit of around ten data points. If there is no satisfactory linear section then try the experiment again.
6. Produce appropriate polynomial curve fits for the other graphs for the same interval of time analyzed on the acceleration graph.
7. Print the graphs as directed above.
Interpretations
1. On each graph write the best fit equation in a more appropriate form using standard symbols for position, velocity, acceleration, and time. (The format of the equation printed by the computer is not readily understandable.) The equations should be of the form: x(t) = , v(t) = , and a(t) = .
2. Determine the derivative of the best fit equation on each position graph and write the result on the corresponding velocity graph using the notation dx/dt = . Write this result immediately below the velocity best fit equation with the terms in the same order for the sake of comparison. The derivative from the position graph should closely match the computer’s best fit equation for velocity.
3. Determine the derivative of the best fit equation on each velocity graph and write the result on the corresponding acceleration graph using the notation dv/dt = . Write this result immediately below the best fit acceleration equation with the terms in the same order for the sake of comparison. The derivative from the velocity graph should closely match the computer’s best fit equation for acceleration.
4. Summarizing the above: each position graph has a handwritten equation x(t) = , each velocity graph has handwritten equations v(t) = and dx/dt = , and each acceleration graph has handwritten equations a(t) = and dv/dt = .
Questions
1. Discuss how well the results support the notion that velocity is the derivative of position and acceleration is the derivative of velocity. Be specific and refer to the graphs and/or equations.
2. Pick the derivative that worst matches the corresponding curve fit equation and determine the percent difference in each corresponding coefficient. This is a little like putting an upper bound on the error in your results.
3. (a) What was the maximum downward acceleration of the balloon according to the data shown on the graph? (b) Determine the percent difference from g. (c) Explain why the balloon does not have an acceleration of g.
4. (a) How does the magnitude of the balloon’s acceleration while it is rising compare to the magnitude of the balloon’s acceleration while it is falling? Which is greater? (b) What is a plausible explanation for this difference?
5. Suppose the balloon had not been caught and continued to fall indefinitely (with no obstacles below it). Do you think the equations that you found for the balloon would accurately describe the balloon’s motion as it continued downward? Explain and be specific.
6. Discuss error. What are the indications of error and likely sources? Be specific.
Your report (30 pts.) shall consist of the following material – neatly labeled and in this order:
q Graph of position vs. time for hand w/ x(t) = (3)
q Graph of velocity vs. time for hand w/ v(t) = and dx/dt = (3)
q Graph of acceleration vs. time for hand w/ a(t) = and dv/dt = (3)
q Graph of position vs. time for balloon w/ x(t) = (3)
q Graph of velocity vs. time for balloon w/ v(t) = and dx/dt = (3)
q Graph of acceleration vs. time for balloon w/ a(t) = and dv/dt = (3)
q Answers to questions 1 – 6 (12)