You are correct that accuracy in measurement will depend on the number of
horizontal lines. This is why I suggested you design the experiment and
analyze only seven or eight data points that fill the vertical range of the
video. In your first message, I thought you were concerned that the velocity
versus time graph was not linear. I do not have that problem.
You may be trying to over-analyze the experiment. If this is what you want
your students to do also, then by all means continue. It depends on your
audience. I approach this experiment with my introductory students in the
following fashion. If your course is an experimental design course, then
approach it more rigorously.
1. Our model is that the acceleration is constant. Therefore, position
versus time and displacement versus time plots should not be linear, but
velocity versus time should be linear. If this holds experimentally, then we
can analyze the velocity versus time graph and find the slope and intercept.
Using these two values, we can write equations for the object's velocity,
displacement, and after measuring the initial position from the position
versus time plot, an equation for position too.
2. Do the experiment, then observe the position versus time and displacement
versus time graphs. Are they linear?
3. Observe the velocity versus time graph. Is it linear? If so, find the
slope and intercept. What are the units of these values? What are the
physical meanings of these values? [Here is where you will have problems if
your experiment is not well designed. If velocity versus time does not look
linear, then hand waving comes in, or a more rigorous analysis as to why it
is not linear must be performed. Again, I don't have this problem and don't
have to hand wave, because the velocity versus time graph does look linear.
Of course, sometimes it looks better with my glasses off.]
4. Using the slope and intercept, write position, displacement, and velocity
equations. Check these equations using their corresponding graphs.
How you have your students analyze error is up to you. This could be as easy
as having them draw three straight lines through the velocity versus time
graph, using the eyeball method to find the best slope (& intercept), the
maximum slope (& intercept), and the minimum slope (& intercept). They can
then write three equations each for position, displacement, and velocity.
However, this may be difficult if the velocity data is good to begin with.
While I stop with the velocity versus time graph, you seem to want to proceed
to the acceleration versus time graph. Why stop there? Why not look at
delta(a)/delta(t) versus time? And the next derivative? My approach is, if
it looks linear, smells linear and tastes linear, then treat it linear and
test it against the data. If it works, then we're done.