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POSITION, VELOCITY, AND ACCELERATION Exp 1
Introduction to the Pasco Interface
(Alward/Harlow web File: “motion.doc” 1-15-04)
Name: ______Partners: ______Section No. ______
Equipment Needed: 2.2 meter track collision cart 3x5 card motion sensor
masking tape level
Note to instructor: Physics 17 and Physics 23 students will skip Part V-D.
I. THEORY AND PURPOSE
In this simple exercise, the student will gain familiarity with the data gathering and analysis capabilities of the Pasco hardware and software. The student will also receive an introduction to the qualitative and quantitative relationships between position and velocity
The Pasco Ultra-Sound Motion Sensor transmits sound at a frequency beyond the range of human hearing; these sound waves are reflected off of an object and received by detector that is packaged together with the transmitter. The round-trip is timed by the computer and the known speed of sound is used to calculate the distance to the reflecting object. The position, velocity, and acceleration of the object are calculated several hundred times each second, and this information may be displayed in tabular or graphic form.
The following equations govern one-dimensional motion (motion along a line):
x = x0 + v0t (if a = 0) (1)
v = v0 + at (if a = constant) (2)
If the acceleration is zero, then v = constant and a graph of velocity versus time is a flat, straight line. If the acceleration is constant, but not zero, then the graph of velocity versus time is straight, sloping upward (for positive accelerations), or sloping downward (for negative accelerations).
In this investigation a cart will be sent at nearly constant velocity down a track, where it will rebound against a magnetic bumper and return back along the track at nearly constant velocity. However, during the brief period of time of the collision with the bumper the cart will experience a non-constant acceleration. In what follows below, the user will measure the velocity and acceleration of the cart during several phases of its journey.
II. SETTING UP THE COMPUTER INTERFACE.
If graphs of position, velocity, and acceleration versus time are already on the computer monitor, you may skip the steps below and go to Part III.
If graphs are not already on the computer monitor,
1. Turn off the computer, then turn on the signal interface, then turn on the computer again.
2. Access the Main window and click the Science Workshop icon.
3. Open the file motion.sws, and select “Don’t Save” if you are invited to save changes to a previous document.
The monitor should now be displaying three graphs, one for position versus time, one for velocity versus time, and the other for acceleration versus time.
III. SET UP TRACK, CART, AND MOTION SENSOR
1. Place the 2.2 meter Pasco track on the lab table, with the 227 cm end next to the wall. If necessary, level the track.
2. Place a bumper at the 200-cm mark. Place a second bumper at the 50-cm mark, with the metallic magnetic disks, which are on the face of the bumper, facing the first bumper. This second bumper will magnetically repel the cart when it approaches.
3. Place the motion sensor next to the bumper, at the 199-cm mark, and facing down the track. To avoid reflections from the track, incline the front of the sensor very slightly by wedging a folded piece of paper or other object between track and the bottom of the sensor.
4. Tape a 3x5 card across the back end of the cart, and place the cart on the track at about 190 cm, with the 3x5 card facing the motion sensor.
IV. COLLECTING MOTION DATA
If the motion sensor cables are not already connected to the interface box (the black box), connect them now: the yellow cable goes to Channel 1, the other one to Channel 2.
Note: the sensor is not capable of reliably sensing signals reflected and received too soon after they were sent. Consequently, you should not attempt to monitor objects which are closer than 0.40 meter from the sensor.
One begins the collection of data in one of two ways: Either hold down the ALT key and then press the R (record) key, or use the mouse to open the "Experiment" window, then click on "Record".
r Gently push the cart; begin collecting data when the 3x5 card passes the 160 cm mark.
Practice launching the cart until you are able to achieve an approximately 3-4 seconds travel time for a round-trip journey. (Set horizontal scale--the time scale--to zero to five seconds, and make sure the clocks stops after four seconds.)
If unexpected spikes or other irregularities appear on your velocity graph, it is probably because the ultra-sound is reflecting off the track or some other nearby object. Graphs similar to the ones shown below should appear on the screen. As soon as you obtain graphs at least as smooth as the ones blow, you may go on to Part V.
V. ANALYZING MOTION DATA
A. Forward Motion: (Almost) Constant Positive Velocity
1. We will first obtain an approximate value for the velocity of the cart during its journey toward the bumper. Place the mouse cursor at the upper left corner of the top graph, then hold and drag it diagonally, then release. A box similar to the one in Region A below will appear.
2. At the bottom left corner of the screen are six icons. The first one, S, allows access to the statistics software. Click on the statistics icon now. Select the options shown in the figure below.
3. Repeat this procedure for Regions B and C.
4. The value of a2 shown in the statistics window in the top box is the slope of the position versus time curve. This slope is the average (mean) value of the cart’s velocity. Report this value in the A box at the top of Table I.
5. The mean value of the velocity is found in the statistics window for the region marked "B". This value should be close to the value of the slope a2 of the position curve in Step 4. Report the mean value (the mean "y" value in the statistics window). If the mean value here is not the same as the mean value obtained by another method in Step 4, why do you think that the two values aren't identical? Give your answer below:
6. Another way to obtain the average velocity of the cart is to calculate the slope of the straight line in Region A by dividing the rise by the run. Do this now by using the "cross-hairs" cursor: click on this icon (the second one, next to the S icon). Move the crosshairs to two widely separated points on the line and record the values of x and t at each point. Use these values to calculate and record the slope. The value you obtain should be close to the values for the velocity you obtained above.
7. During the forward journey, the cart’s velocity was almost constant. Friction will cause the cart velocity to decrease slowly; for that reason, the position vs time curve is not perfectly straight (which it would be if the velocity were constant), the velocity curve is not perfectly flat (which would mean that the velocity is constant), and the acceleration data in region C is not exactly zero (which it would be if the velocity were not decreasing slightly).
Read and record the value of the slope of the velocity curve in region B; it will probably be small, because the velocity is changing slowly. This slope is the average acceleration of the cart. Why is the slope negative, not positive? Give your explanation below:
8. Compare the value of the acceleration found in Step 7 to the mean value of the acceleration found from the third curve (Region C). Report this mean value in Table I . The two acceleration values should both be small; ideally, they both would be zero. No numerical comparison is required here; just observe the values.
Table I: Forward Motion at Nearly Constant Velocity. Velocity Calculated Three Ways
A / Slope a2 = m/s (from statistics window)(Slope of line in Box A; use linear fit data)
B / Mean velocity = m/s (from statistics using "mean" value)
(Mean value of velocities in Box B)
A / Crosshairs Measurement of Slope:
A / x1 = m t1 = s Data points on the line in Box A.
To be used for rise and run
x2 = m t2 = s calculation.
A / Rise = x2 - x1 = m
A / Run = t2 - t1 = s
A / Slope = Rise/Run = m/s
(This is the slope of the line in Box A; it is the average velocity of the cart. This value should be close the two velocity values at the top of this table.)
B / Slope of velocity curve in Box B = m/s2
(Use linear fit data)
C / Mean acceleration = m/s2
Taken from Box C; the acceleration, ideally, is zero.
This value should be close to the value just above.
B. Average Acceleration During the Collision
1. Prior to collision with the bumper, the velocity of the cart was positive. Afterward, it is negative. This is shown in Region D in the figure below. Box this region using the method of click, holding, and diagonally dragging the cursor across this region of your graph.
The slope of the velocity versus time curve is the average value of the acceleration during this period. Use the statistics window to obtain the value of the slope of the line (a2) which is linearly fitted to this data. This value should be negative, because cart is losing velocity during the collision. Report the value of a2 in Table II.
2. Find the mean value of the acceration in Region E. Use the same box width and data points as were used in Step 1 above. The mean value obtained here won't be the same as found from the slope a2 of the velocity curve, because the curve fit is only approximate. The purpose of these steps is just familiarize you with motion curves, not obtain exact measurements.
Table II: Acceleration During Collision (Calculated Two Ways)
D / Slope a2 = ______m/s2(The slope of the line in Box D)
E / Mean acceleration = ______m/s2
(Should be close the acceleration value found above.)
C. Reverse Motion: (Almost) Constant Negative Velocity
1. Select the regions shown in F, G, and H, and record the values of the position vs time slope (a2), and compare it to the mean value of the velocity in Region G. A quantitative comparison is not required here; just observe that the values are approximately equal. Note also that the velocities are negative, as expected, since the position curve is sloping downward (and also because the cart is returning home.
2. Likewise, compare the value of the slope (a2) of the velocity vs time graph in Region G to mean value of the acceleration in Region H.
Table III. Reverse Motion
Average (mean) Velocity During Reverse Motion (Calculated Two Ways)F / Slope a2 = m/s
G / Mean velocity = m/s
Acceleration During Reverse Motion (Calculated Two Ways)
G / Slope a2 = m/s2
H / Mean acceleration = m/s2
D. Displacement = Integral of Velocity-Time Curve
(Note to instructor: APS 17 and APS 23 students will skip this part.)
The displacement of an object during some time interval is the net distance traveled during that time interval. For example, the displacement of our cart during the entire time period of the journey is approximately zero, since the final and initial coordinates are about the same.
During an infinitesimally small time interval, dt, Equation 1 shows that the infinitesimal displacement, dx, is given by dx = vdt, where v is the average velocity during the infinitesimally small time interval dt centered on some instant of time, t, (this velocity is usually called the instantaneous velocity).
If the velocity is constant over some time period Dt (not necessarily a small time period), then the displacement during this interval is given by Dx = vDt.
But, if the velocity is not constant, then the displacement is given by the infinite sum of products:
Dx = v1Dt + v2Dt + v3Dt + ......
or, Dx = òvdt
which is just the "area" under the velocity curve during the time interval in question.
Thus, the final value of the position coordinate, x, is equal to the initial value, x0, plus the increment, Dx.
x = x0 + Dx
or, x = x0 + òvdt
1. Remove the acceleration graph by clicking near the small black inverted triangle in the box on the left side of the acceleration graph. Select "DELETE INPUT".
2. Click-hold-drag a box around the data as shown on the velocity curve below.
3. Select the statistics option and integrate the velocity curve. Report the integral value in the table.
4. Use the crosshairs cursor to read the values of x and x0 corresponding to the edges of the shaded region. Record these values in the table.
5. Use the value of the integral to calculate the predicted value of the final coordinate, x. Compare this value to the one found in Step 4.
6. Roughly estimate the value of the integral by estimating the areas of the regions I and J. The estimation is done by multiplying the average width of the region times the average height, both of which are estimated by eye. Note: the area of I is positive, while the area of J is negative. Add these two areas and record the value in the table. It should be within 20% of the value recorded from Step 3. The purpose of this step is to emphasize the fact that an integral is nothing more than an area. In this case, the integral is is an area in velocity-time space.
Table IV. Integral of Velocity-Time Curve
I / x0 = m (from crosshairs measurement)
J / x = m (from crosshairs measurement)
IJ / x = x0 + Dx = m (Compare to value of x above)
I / Area of shape I = m2
J / Area of shape J = m2 (this area is negative)
IJ / Sum of areas above = m2 (Compare to value of Dx)