Determining uncertainty for coefficient of restitution (COR) of a bouncing ball

Question

1. Here's a description of how I calculated COR:

   • Formula $COR=√(h/H)$
   • I have 7 intervals where I measured $h$ and $H$
   • I did 5 trials for measuring $h$ and $H$ for each interval (in total 35 measurements for each of them)
   • The absolute uncertainty is the same for both $h$ and $H$ and is 0.002 m

2. My question is:

   • How do I propagate the uncertainty for the COR?

Answer 1

You have values for 7 different heights. For each height you can compute the error using standard methods. Combining the 7 values you want to take the weighted mean of the 7 CORs. Do you know how to compute the error for a weighted mean?

Answer 2

When doing a statistical analysis you can ignore the absolute uncertainty in the measurements, particularly when it is the same for each.

I would plot a graph of $h$ vs $H$, either using all 35 data points or using the 7 average values of $h$ if the values of $H$ were all the same. This will show you if the COR is constant over the range of your experiment. If it is not changing, you can fit a trend line of the form $y=bx$; the slope of this line is $b=e^2$ where $e$ is COR. You can use the squared deviations of data points from this line to calculate the Standard Error in the slope. If the slope is changing you can fit a quadratic $y=bx+cx^2$ then again use the value of $b$, which is the slope at small values of $x$.

You can do all this in a spreadsheet. Excel has built-in statistical tools which can help you. See http://www.chem.mtu.edu/~fmorriso/cm3215/UncertaintySlopeInterceptOfLeastSquaresFit.pdf. Or you could use data analysis software such as Origin or CurveFit to get the Standard Error.

When you have the Standard Error $S_b$ in slope $b$ then the COR is given by
$e^2=b\pm S_b=b(1\pm\frac{S_b}{b})$
$e=\sqrt{b}(1\pm \frac{S_b}{b})^{\frac12}\approx\sqrt{b}\pm\frac{S_b}{2\sqrt{b}}$.

If the above is too much effort, calculate the COR for each of the 35 readings, then use the data analysis tools in a spreadsheet to calculate the mean $m$ and standard deviation $\sigma$. Even basic models have AVERAGE and STDEV. The Standard Error in the mean is $S_m=\frac{\sigma}{\sqrt{N}}$. Your result is $m\pm S_m$.