Concept: linear motion
Time: 30 m
SW Interface: 500 & 700
MacintoshÆ file: P05 Position, Velocity, Acceleration
WindowsÆ file: P05_POSI.SWS
Science Workshopô Interface
string
motion sensor
track, 2.2 meter (optional)
fan cart
The purpose of this laboratory activity is to study the relationship between position, velocity, and acceleration in linear motion.
The equation for position (x), given a constant acceleration (a), is
where xo and vo are the initial position and velocity. Note that if xo and vo equal zero, the equation is a parabola.
The first time derivative (which corresponds to the slope of the graph) of this equation,
is the well-known equation for velocity under constant acceleration. This equation is linear; the slope of the line is the acceleration.
The second time derivative is the constant acceleration.
For this activity, use a motion sensor to meaure the position of a fan cart as it moves away from the motions sensor. The Science Workshop program will plot the cart's position (x), velocity (v), and acceleration (a).
1. Connect the Science Workshop interface to the computer, turn on the interface, and turn on the computer.
2. Connect the motion sensor's phone plugs to Digital Channels 1 and 2 on the interface. Plug the yellow-banded (pulse) plug into Digital Channel 1 and the second plug (echo) into Digital Channel 2.
3. Open the Science Workshop file titled
as shown:
Macintosh: P05 Position, Velocity, Acceleration
Windows: P05_POSI.SWS
The document has a Graph display of Position (m), Velocity (m/sec), and Acceleration (m/sec/sec) versus Time (sec).
(Note: For quick reference, see the Experiment Notes window. To bring a display to the top, click on its window or select the name of the display from the list at the end of the Display menu. Change the Experiment Setup window by clicking on the "Zoom" box or the Restore button in the upper right hand corner of that window.)
You do not need to calibrate the Motion Sensor.
1. Place the motion sensor and fan cart on
a flat horizontal surface. Make sure that there is nothing to
interfere with the motion sensor's view of the fan cart. (If you
have a PASCO dynamics track available, use it.)
2. Tie a string to the fan cart so that you can hold the fan cart before releasing it without interfering with the motion sensor.
1. Hold the string so that the fan cart remains stationary about 40 cm in front of the motion sensor. Turn the fan on.
Make sure that the fan cart is pulling away from the motion sensor!
2. Click the "REC" button to begin data recording.
3. Release the fan cart.
4. When the cart has traveled the length of the space available, (generally less than two meters is best), click the "STOP" button to end data recording.
"Run #1" will appear in the Data list in the Experiment Setup window.
1. Select the Graph display to make it active. Select "Save AsÖ" from the File menu to save your data. Optional: If a printer is available, select "Print Active Display" from the File menu.
2. Click the "Statistics" button in the lower left corner of the Graph to open the Statistics area. Click the "Autoscale" button to resize the Graph to fit the data.
3. Click the "Magnifier" button. The cursor changes to a magnifying glass shape. Use the cursor to click-and-draw a rectangle around a relatively small region of the plot of position versus time that shows a smooth parabolic curve. Try to surround about five or six data points with your rectangle.
The Graph will rescale to fit the region you selected. (The velocity and acceleration plots will change as well.)
4. Use the mouse to click-and-draw a rectangle around only two consecutive points of the position curve.
5. Click the "Statistics Menu" button in the Statistics area of the position versus time plot. Select "Curve Fit, Linear Fit" from the Statistics menu.
A "best fit" line will be drawn through the two points in your selected region. The coefficient "a2" in the Statistics area is the slope of the line through the points.
6. Click the "Smart Cursor" button. The cursor changes to a cross-hair. Move the cursor/cross-hair to the mipoint of the line segment joining the two points in the region you selected for the position curve.
The coordinates of the cursor are shown next to the vertical and horizontal axes.
7. Compare the slope given in the position plot Statistics area with the value of velocity at that same time. Hold down the Shift key and move the cursor vertically into the plot of velocity versus time.
The velocity at that point is the Y-coordinate shown in the area next to the vertical axis.
8. Repeat the data analysis process for several points on the position plot.
In the plot of position, click anywhere outside your small rectangle to clear the selection.
Click the Autoscale button to rescale the Graph to fit the data.
In the Statistics area of the position plot, click the Statistics Menu button and select "Curve Fit, Polynomial Fit" in the Statistics menu.
In the Statistics area of the velocity plot, click the Statistics Menu button and select "Curve Fit, Linear Fit" in the Statistics menu.
In theStatistics area of the acceleration plot, click the Statistics Menu button and select "Curve Fit, Linear Fit" in the Statistics menu.
In the plots of position and velocity, the "chi^2" value indicates how well the data fit the selected curves.
In the acceleration plot, examine the value of "a2", the slope of the line of "best fit".
1. What are the appropriate units (in MKS)
for the slopes of the position and velocity plots?
2. Why is the acceleration plot so much "noisier" than the other plots?
(Hint: Use the analysis method in Analyzing
the Data to compare the slope of the line segment between any
two points on the velocity plot with the value on the acceleration
plot that corresponds to the midpoint on the line segment in the
velocity plot.)
1. How close does the plot of position
fit a polynomial curve? (Remember, the closer "chi^2"
is to zero, the better the fit of data to the curve.)
2. How close does the plot of velocity
fit a linear curve?
3. Is the acceleration nearly constant? (Remember, a slope of zero indicates a "constant" value.)