Purpose: Given the essential
equipment, you and your partners will have to determine methods of measuring
speed as precisely and with as little uncertainty as possible. The focus is on data analysis, particularly
on methods to determine uncertainties and propagate them to results.
Materials: Marble Ramp Meter Stick Stopwatch Graph
paper
Procedure:
You are given the task of trying to
measure the speed (distance / time) of a marble that rolls off a ramp. You want to try and maximize accuracy and
precision while limiting uncertainties.
As far as procedure, write detailed steps you decide on for a method
that involves extracting a speed from
a graph.
Do not expect much help from me as
far as measurements. I suggest that you
take a few minutes before starting to write down possible uncertainties (experimental,
which you may be able to minimize) and then give this a try.
Data Table: Develop a neat, organized,
and appropriate data table for all measurements for your procedure. Your group will need to take some number of
trials for time measurements for each distance that you use. In your data table(s), include the standard deviation for each set of time measurements.
s_{time} = æ [S(t_{avg} – t_{i
})^{2}] ö ^{½}
è N – 1 ø
Questions/Analysis:
1. What potential experimental
errors did you and your partner(s) decide were the most significant? Explain why these caused the most worry. Do not just say something like “human error”
or “calculation error;” but be detailed and thorough.
2. Is it possible to “minimize”
errors in this experiment? Why or why
not? (Think about your equipment) What did you try in order to minimize
uncertainty on time measurements?
3. Why is it important to be
able to accurately determine
speeds? Think in terms of everyday
events or situations; try to come up with two examples.
4. Why is it important to be
able to precisely determine speeds
(i.e. measure down to more decimal places)?
Think in terms of everyday events or situations; try to come up with two
examples.
5. Make a graph of distance (y)
versus time (x) using Excel. Find the bestfit line and R^{2} value
for your data. Then, put on error bars
in the time dimension (this will likely have to be done by hand after printing
out your graph. Don’t forget to label
axes, include units on the axes, and so on, when making a graph). The error bars for time will make use of the
standard deviation for that particular set of time trials. It is OK for different points on a graph to
have different sized error bars; in fact, you should expect to have different
sized error bars for each data point.
6. If you started your marble
at the same height on the ramp each time you can assume it is going the same
speed when it rolls off on a flat surface.
When you made your graph of distance versus time, this is why you would
expect a straight line. Are all your
points on the line? Don’t feel bad if
they are not; rarely will data lie on the same line. Determine
and report the “best” speed of your marble as well as estimates of uncertainties on your speed result from your graph,
using max and min slope lines as determined by your error bars. In class, you will see an example of this.
7. Calculate the uncertainty for your speeds by
using the quadrature method for
independent measurements (it is safe to
assume that distance and time measurements
are independent of each other and random
since you’re using different measuring
devices for each). Do
this for each individual point of
distance and time. How do
these uncertainties compare to those from
your graph, using the max and min lines?
dv = v
[(dx/x)^{2}
+ (dt/t)^{2
}] ^{½} , where dt is
the standard deviation for that set
of times, and dx = 1 cm = 0.01 m.
Note that dx/x
and dt/t
are called the fractional uncertainties of the distance and time
measurements.
We are adding the
fractional uncertainties in quadrature (looks a lot like Pythagorean theorem!).
Grading:
Title (always have this) 1
point
Purpose (always have this) 1
point
Materials and Methods (Procedure) 3 points

procedures should be written such that someone who has not done this before can
recreate your lab
Data and Graph 5
points
Analysis Questions 10
points
Above and Beyond: If interested in learning the theory behind leastsquares fits to linear data, look here.
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