Q: How the data or models given in physics or mathematics problems can be obtained in real life? When ever there is physics or mathematics problem. we are already given a mathematical model. Can't we make own models from given diagrams or data collected from the problem? sometimes angle is given in the problems like a plane makes an angle of theta= pi radian etc... how they measure angle of a plne and how can we measure angle in real life... suppose if I want to calculate the length of a ladder that is leaning against a wall & ladder makes an angle with respect to ground , how that angle can be measured..? .. in physics books if we want to find position by integration we already are given acceleration as a function of time..why is it so..? can't we obtained these functions; models; or angles etc ourself?

A: There are essentially two ways in which we obtain functions and models ourselves: through the use of an appropriate measuring instrument (ruler, protractor, scale, clock...) or via mathematical logic.
Measuring instruments can give us an excellent but imperfect approximation, while mathematical logic provides us an exact value based upon input values that are either theoretical or approximate.
Most school mathematics assumes values that are exact in order to encourage us to work through the mathematical logic. In practice, however, we may be working with measurements that rather poorly known (especially in sciences such as astronomy). For this reason, we calculate the uncertainty of our results based upon how well we were able to make our measurements.
