BHS -> Staff -> Mr. Stanbrough -> AP Physics-> Kinematics-> this page
Undoubtedly, you have used a motion sensor and computer interface in other classes to plot graphs of position, velocity, and acceleration versus time, but how does the "magic box" do its "magic"? In physics, we don't want to treat laboratory equipment as "magic boxes" any more than we have to, and calculations of position, velocity, and acceleration are certainly something that we should be able to understand.
The motion sensor sends out a series of ultrasonic pulses. These pulses reflect from nearby objects and return to a detector. The computer software provides a very precise timer to measure the time elapsed between the sending of the original pulse and the detection of the reflected pulse.
The pulses travel at the speed of sound in air, v (about 345 m/s, depending on atmospheric conditions). Suppose that the pulses reflect off an object that is a distance d from the motion sensor, and that the reflected pulse returns to the detector after time t. It's a simple application of "distance equals rate times time", except that the distance traveled by the pulse is 2d:
2d = vt, so
Your velocity is the slope (derivative) of your position vs. time graph, so if you were at position d1 at time t1, and position d2 at time t2, your velocity can be approximated as:
(Equation 2)
In the same way, your acceleration is the slope (derivative) of your velocity vs. time graph, so if you have velocity v1 at time t1, and velocity v2 at time t2, your acceleration will be approximately:
(Equation 3)
Pasco Science WorkshopTM interface |
Motion Detector |
air track & glider or dynamics track (optional) |
You can set up the motion detector to measure a motion that you create, or the motion of a glider on a sloping air track - whatever. The actual motion isn't the point here, just its analysis.
Hint:Turn on "Show Balloons" in the Help Menu. It will explain the purpose of the various icons in the Pasco interface - which is a big help.
Record a motion or two - we just need some data to look at.
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Compare this formula to equation 1 above. The default speed of sound (344 m/s - If you calibrated the motion sensor to some other value, use it here. Did you forget it? Double-click the motion sensor icon ()) was used in this calculation, and the "@1.tEcho" comes from the "Input" menu. It would be more efficient to use "172*@1.tEcho", but what I have looks more like equation 1. |
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Here is the calculation for the velocity. The "derivative" comes from the f(x) - Special menu, and the myx comes from the "INPUT" menu. The "4" means that the program calculates the slope between the current time and 4 time intervals later. |
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The calculation for acceleration is shown above. Why "2" here and "4" above? I don't know - but it matches the Science WorkshopTM calculation though, doesn't it? |
Now that you know how the motion sensor determines your position, velocity, and acceleration, you can go ahead and use the built-in calculations when you need them, you don't have to create your own calculations each time. I'm sure that you've noticed that the kinematics graphs produced by the motion sensor are sometimes (ok, often - well, generally) pretty rough - especially the acceleration vs. time graphs. This is because slight irregularities in the round-trip times show up as irregularities in the position data, which are magnified in the velocity calculation, then magnified again in the acceleration calculation.
What can you do? First, you should make sure that the motion detector is not getting reflections from some unwanted object. Try clearing the area, re-aiming the detector, and adjusting the beam (near/far). This probably won't remove all of the "jitters" in the data, but it can help a lot. Beyond that, you can either work around it, or try to fix it.
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To create acceleration vs. time graph shown above:
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The Experiment Calculator window shown above displays a smoothing function applied to motion sensor position data. The "smooth" function is found in the "Special" submenu of the "f(x)" menu, and the "@1.x" is found in the "INPUT" menu. |
Well, doesn't that change the data? Yes, it does. Isn't that somehow underhanded and unsavory, if not downright illegal? Well, no, it isn't. If the "jumps and jiggles" are truly random noise (just about the same number of high and low jumps, etc.), then there is nothing wrong with trying to remove it. Just be sure to:
Science WorkshopTM makes it easy to smooth a data set. In the "Special" submenu of the "f(x)" menu in the Experiment Calculator window you will find the "smooth(n, x)" function. Here, "x" is the data set to smooth, and "n" is the number of points to use in the smoothing operation. How many points should you use? You need to play around with it. Too few, and the data isn't really affected by the smoothing operation, and too many can remove a great deal of the information content. It's a judgment call.
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This graph uses the same data as the graph above, but the original round-trip time data (only) was moderately smoothed (n = 5). |
Now, modify your velocity calculation to use the smoothed position
values instead of the raw data, and compare. You can smooth the new
velocity data, too. Now modify your acceleration calculation to use
the smoothed velocity data, etc. (That's a lot of smoothing -
it is possible to "go nuts" with this smoothing thing, you know...)
What do you think?
Hopefully, this lab exercise has given you an understanding of how the motion detector does its detecting, as well as some steps to take to improve the process that you can apply in your future work.