# Physics with Calculus/Mechanics/Measurement

Measurement is one of the fundamental concepts in experimental sciences, including physics. What is exactly meant by "measurement" is not precisely defined, and in fact it is one of the frontiers of research in quantum physics, where the act of measurement plays an important role in the commonly accepted interpretations of quantum mechanics. For our purposes, we can say measurement is the act of writing down a physical quantity, such as time or length, in objective terms.
Measurement is how one tests theories in experimental sciences as physics. In order to test a theory, one uses the theory to make predictions in a specific physical situation, and then perform an experiment, that is, a series of measurements, to see if the prediction is true. For example, the Newtonian theory of gravity predicts as a consequence of Earth's gravity near its surface that an object in free fall travels a distance D in time T, so that D is proportional to T$^2$. Then we devise an arrangement of equipment and a plan to test this prediction: we can drop a steel marble from height of 25 cm, 50 cm, 75 cm, 1.0 m, 1.25 m and 1.5 m, and measure how long it takes to fall that distance using a stopwatch. If our measurements roughly indicate D ∝ T$^2$, then this is a confirmation of the theory in this particular situation. To be more specific, our observations do not contradict the prediction of the theory, and so we can assume it holds in all cases (unless otherwise shown by further experimental data) via an argument such as Occam's razor.

## Physical Quantities

A physical quantity is anything that we use to describe a physical phenomenon quantitatively. We used a few examples of physical quantities earlier without discussing them further: distance, time, area, and mass. And we will make what we know of those quantities from intuition more precise here.
Some physical quantities can be defined from other quantities. Area and volume are good examples: when we have a square and a cube we can define their respective area and volume in terms of the length of the sides. Such quantitative definition also tells us how to calculate that particular physical quantity, given some other physical quantities.
Intuition tells us that we can't define all physical quantities in such manner, not without making a circular definition that will not be very useful to us. Physical quantities that cannot be defined in terms of other physical quantities are considered to be fundamental and we can define them only by stating how we measure them. Such definition is called operational definition.
Distance, time, and mass are such fundamental physical quantities. These cannot be defined in terms of other measurable quantities (note that there is some arbitrariness in this choice: it is possible, although not practical, to choose area to be a fundamental physical quantity and define distance as length of the side of a square having certain area). In fact, how we choose to define them determines our unit system, which is a topic of next chapter.
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