Why do we use a second order polynomial to fit our position versus time data?

Why do we use a second order polynomial to fit our position versus time data? Please don't say because it looks good or fits the data. Give a physics-related explanation.
Express your experimental value of g (acceleration due to gravity) using a 95% confidence interval and with correct significant figures.
Does the true value of g=9.81 m/s2 lie within your range? The idea of a 95% confidence interval is that there is a 95% chance that the true mean lies within that range.
Would you expect the data to fit a polynomial function as well if the ball was either much less dense or was allowed to fall for a longer time (and therefore much faster)? Please explain.
Should the value of gravitational acceleration from your data depend on the time starting from zero or the ball starting from rest? Please explain carefully.