Linear Motion with Constant Acceleration

Transcription

1 Linear Motion 1 Linear Motion with Constant Acceleration Overview: First you will attempt to walk backward with a constant acceleration, monitoring your motion with the ultrasonic motion detector. Then you will use photogates to study the motion of a hotwheels car rolling down a straight inclined track, to determine whether its acceleration is constant. Physics principles: One-dimensional motion Acceleration Motion with constant acceleration New lab skills: Manipulating data to obtain a linear graph Measuring the slope of a graph and its uncertainty Equipment needed: Straight inclined track and toy car Computer and ScienceWorkshop interface Meter stick Two photogates and stands Motion detector and stand Instructions Part 1: Moving at Constant Acceleration In this part of the experiment you will attempt to walk backward with a constant acceleration, monitoring your motion with the ultrasonic motion detector. Open the Constant Acceleration document in the Lab Files folder on your computer s desktop. Connect the motion detector in the same manner you done in the previous lab (yellow plug in port 1, black plug in port 2). If you need assistance, you might review your previous lab s instructions or consult your instructor. Stand about half a meter in front of the motion detector. Click the start button and start moving away from the detector. The idea is to move such that your acceleration remains constant, that is, the velocity, which is displayed on the computer screen, is a straight line sloping upward. You ll have to start very slowly and then uniformly increase the speed of your motion.

2 Linear Motion 2 Figure 1: Toy car moving between photogates. This is a challenging exercise and you may need to practice a few times before you get a feel for motion with constant acceleration. You may have to adjust the scale of the time and velocity axes so the graph fills most of the screen. Each member of your group should do this exercise separately, and make a separate printout of the motion graph. When you get your printout, mark the section of the graph that represents approximately constant acceleration. Use a ruler to draw a straight line that approximates this section of the graph. Measure the rise and run of this line, and use them to calculate the slope of the line, which is your approximate acceleration. Show your arithmetic and your result directly on the printout. After everyone is finished with the motion detector, unplug it from the interface box. Part 2: Motion of a Car on an Incline For the rest of this lab session you will study the motion of a toy car rolling down a straight inclined track. Collecting data. Connect the photogates to the interface box with the first photogate that the car encounters plugged into port 1, and the photogate the car passes through second plugged into port 2. Open the Linear Motion document in the Lab Files folder on the computer s desktop. This program will measure three time intervals: a short time interval t 1 (the time it takes for the car s flag to cross photogate 1); a longer time t (the time interval from when the flag enters photogate 1 to when it enters photogate 2); and another short time interval t 2 (the time it takes for the flag to cross photogate 2). Recall how to calculate instantaneous velocities using the short time intervals t 1 and t 2, along with a specifically measured distance. (If you cannot remember how to do this, refer to last week s lab or ask your friendly lab instructor for hints). These velocities will be referred to as v 1 and v 2 and will be used in equations later in this lab. Position photogate 1 about 7 cm down from the top of the track, at a height such that the middle of the car s flag will block the beam. Place photogate 2 exactly 10 cm farther down the track, also at an appropriate height. We ll use the symbol x for the separation distance between the photogates. Press the Start button on the screen, then release the car from the top of the track. You should see numbers appear in the first row of the Table. Leave the program running and move photogate 2 down to a distance of x = 20 cm from photogate 1, then release the car from the top again. Repeat for x = 30, 40, 50, and 60 cm.

3 Linear Motion 3 To analyze the data you ll use the Excel spreadsheet program. Click the Excel icon in the dock to start the program, then copy the data from Table t (Timer 2) of the timing program into the Excel spreadsheet, starting at cell A3 (you should have six rows, two columns of copied data). In column C enter the corresponding distances, 10 through 60 (in centimeters). You may also want to copy data from Tables t 1 and t 2 (Timers 1 and 3) somewhere onto the spreadsheet in order to calculate averages for v 1 and v 2 ( =average() ) as well as standard deviations ( =stdevp() ). Enter appropriate headings (including units) for each column of the spreadsheet, and enter your names in cell A1. Position versus time: Make a plot of the position (separation distance x, on the vertical axis) versus time (elapsed time t, on the horizontal axis). Use the ChartWizard tool as in the previous lab, choosing an XY (Scatter) plot with gridlines. Enter Figure 1 Distance vs. Time for the chart title, and enter appropriate labels, including units, for both axes. Move and/or resize the graph as necessary so that it fills a page, and print copies of this page for each group member; do not print the data table yet. (Before printing use Print Preview under the File menu to check what the printout will look like.) Carefully draw a best-fit smooth curve through the data points on your graph. Then go to the Report page and answer all parts of Question 1. Linearizing the data. Your next task will be to use this data to test whether the acceleration of the car is constant, and to extract a best value for its acceleration. If the acceleration is constant, then the position of the car as a function of time should be given by the equation x = x 0 + v 0 t at2, (1) where x 0 and v 0 are the car s position and velocity when t = 0. Since we re using t for the time to go from one photogate to the other, time t = 0 is when the car is at the first photogate (not when you released it); therefore x 0 is zero and v 0 is the same as what we ve called v 1 : x = v 1 t at2. (2) This equation predicts a specific shape for the graph of x vs. t. But because this shape is not a straight line, it s hard to tell from a graph of x vs. t whether the equation is valid. In such circumstances, it is standard practice to manipulate the equation so that the left-hand side is a linear function of the independent variable (t). One way to do this is to divide through by t: x t = v at. (3) 2 This equation is a linear one, of the form y = mx + b. That is, if we plot x/t vertically vs. t horizontally, we expect to get a straight line whose slope (analogous to m) is a/2, and whose intercept with the vertical axis (analogous to b) is v 1. If we don t get a straight line, we can conclude that the equation does not apply, in other words, the acceleration isn t really constant. In column E of your data table, enter appropriate formulas to calculate x/t for each of the six rows. (You can save time with the Fill Down command under the Edit menu.) Make a plot of x/t versus t, and call it Figure 2 x/t vs. t. Enlarge the graph to fill an

4 Linear Motion 4 Figure 2: Geometry for determining the track angle. entire page and print a copy of this page for each person as before. Answer Question 2 in the Report. To extract a value for the car s acceleration you can draw a straight line through the data points on your Figure 2 and measure the slope of this line. However, it s not enough to just get a single value. What we want is a range of values, indicating a range of uncertainty for the true acceleration. Therefore, use a ruler to carefully draw two straight lines through the data: one that is the steepest plausible line that still closely represents the data, and one that is the shallowest plausible line. Each line should come fairly close to all six data points. Carefully determine the slopes of both lines, showing how you did so right on the graph. Use these slopes to compute two values for the car s acceleration (remember that the slope is not equal to the acceleration; refer back to equation 3). The average of these two numbers would be your best value for a, while the difference between the average and either of them would be the uncertainty. Enter these results in the Report. Velocity versus time. The velocity of an object moving with constant acceleration obeys the equation v = v 0 + at, (4) or in our notation, v 2 = v 1 + at. (5) Because you ve measured v 1 and v 2 directly, you can use this equation to again test whether a is constant and to extract a value for a. This time the equation is already linear, so no further manipulation is necessary. Make a plot of v 2 vs. t, call it Figure 3 Velocity vs. Time, and print copies for each person as before. Draw the steepest and shallowest plausible lines through the data, measure their slopes, and use these to obtain another value for the acceleration and its uncertainty. Enter these values in the Report, and answer Questions 3, 4, and 5. Theoretical acceleration. Now that you have compared your two measured values with each other, let us compare them to a theoretical prediction. Using Newton s laws of motion one can easily show that for an object sliding down an inclined plane without friction the acceleration should be a theory = g sin θ, where θ is the angle of the incline (measured from horizontal) and g is the freefall acceleration, 9.80 m/s 2. To determine θ, measure the dimensions of the track apparatus, draw a picture, and use an appropriate trigonometric function. Enter this information in the Report, and calculate the theoretical acceleration. Answer Question 6.

5 Linear Motion 5 Average velocity. The average velocity of an object is defined as the total distance traveled divided by the time elapsed. For the time interval t in our experiment, v average x t = x 0 t 0 = x t. (6) If (and only if) the acceleration is constant, then this is the same as the average of the initial and final velocities: v average = v 1 + v 2 2 (constant acceleration). (7) You ve already computed the average velocity in column E of your spreadsheet. Now, in column F, enter the appropriate formulas to compute the average of the initial and final velocities, (v 1 + v 2 )/2. In column G, compute the percent difference between columns E and F, that is, (one value) (other value) percent difference =. (8) (either value) (When the percent difference is small, as it should be here, it doesn t matter which value you put in the denominator. When the percent difference is large there is sometimes a good reason to use one value, or the other, or the average.) Print a copy of the entire data table for each member of your group, and answer Questions 7 and 8.

6 Linear Motion 6 Report: Linear Motion Name Partners Lab Station Date 1. On a graph of position vs. time, the instantaneous velocity at any time is represented by the slope of the tangent line. With this in mind, answer the following questions about your Figure 1: (a) Take a ruler and (without drawing a line) examine the slope of the line tangent to the curve at several different points. As time increases, does the slope increase, decrease, or remain constant? What does this say about the velocity of the car? What does it say about the acceleration of the car? (b) Just by inspecting the slope of your Figure 1, can you determine whether or not the car has a constant acceleration? (Explain very briefly.) (c) Draw a tangent line (at least 2 or 3 inches long) and its corresponding triangle on your Figure 1 at the point x = 50 cm. Measure the rise and the run of the triangle in the units of the graph, and write these numbers along the legs of the triangle. Then compute the slope, which should be the car s velocity at x = 50 cm. slope at x = 50 cm =

7 Linear Motion 7 (d) Compare this value with the velocity from your table of data; are the values reasonably close? v 2 at x = 50 cm = (e) Estimate the slope of your Figure 1 at the point x = 0 cm: slope at x = 0 = 2. From your Figure 2, does it appear that the acceleration of the car was constant? Be sure to discuss both the random fluctuations in the data, and whether there are any overall trends. Value of a and uncertainty from Fig. 2 = Value of a and uncertainty from Fig. 3 = 3. Are your two values of a in agreement with each other? Explain briefly. 4. Make a rough sketch of what Fig. 3 would look like if the acceleration were increasing with time, and comment on what makes it different.

8 Linear Motion 8 5. The intercept of a graph is the point where the line crosses the vertical axis. For both Figure 2 and Figure 3, estimate the average intercept of your two lines. What physical quantity does this number represent? Discuss whether your intercept values are what you would expect, based on your data and the equations above. Average intercept from Fig. 2 = Average intercept from Fig. 3 = Discussion: Track angle θ = Diagram and calculations: Theoretical acceleration = 6. Are your experimental measurements of a in agreement with the theoretical value? Explain how you can tell, and discuss whether you might expect any discrepancy.

9 Linear Motion 9 7. Discuss the results in the last three columns of your table. What pattern do you see? What can you conclude about the motion of the car? 8. Describe an example of a type of motion for which the formula v average = (v 1 + v 2 )/2 would not apply. (Be as specific as you can.) At the end of this report, please attach your motion detector graph, then a printout of your data table, and finally your Figures 1, 2, and 3.