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Uncertainties in the velocity estimates

 

If we say that the uncertainty of the actual location of what we see when we see a pixel is tex2html_wrap_inline4647 pixel in each direction, then the uncertainty of the location of a particle (track) must be in the range tex2html_wrap_inline4649, where N denotes the number of pixels that make up the image, measured in pixel sizes. The lower boundary comes from the case where all pixels are weighed equally in the average, and the upper boundary comes from the case where a single pixel dominates the average because it has a much higher intensity than the other pixels in the image of the particle.

When we calculate the particle velocity from the difference between two particle positions, as in the Faraday experiment, the upper limit on the uncertainty of the velocity is the equivalent of tex2html_wrap_inline4653 pixel along each of the two axes.

When we estimate the particle velocity from a single image of a particle track, the uncertainty can be larger than the tex2html_wrap_inline4653 pixel we get if we just look at the distance travelled as the distance between the two pixels that are furthest away from each other. This problem will usually occur when the particle is moving very slowly, so the track almost appears as a circle, and the particle just as well could have been moving perpendicular to the direction found by looking for the pixels that are furthest away from each other. This was not a problem with the data collected from the soap film experiment, so I consider tex2html_wrap_inline4653 pixel to be a good estimate of the uncertainty of the velocities measured in the soap film experiment.

When determining the velocity and the uncertainty on the velocity it is very important to use the right unitsgif.

Since the transformation between physical units and camera units can change from measurement to measurement, the uncertainty on the velocity (in physical units) can change between the different measurements as well.

  table1033
Table 4.1: Typical magnitudes of uncertainties in the present experiments. The x- and y-axes refer to the coordinate systems of the cameras.


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