Basics


Identifying Patterns and Regularities

When we observe a particular phenomenon over and over again, we begin to get a sense of how nature behaves. We start to recognize patterns in nature. Eventually, we will generalize our experience into a synthesis that summarizes what we have learned about the way the world works.
Scientists often summarize the results of their observations in a mathematical form, particularly if they have been making quantitative measurements. In the case of a falling object , they might measure the time it takes an object to fall a certain distance, rather than just noticing that the object falls.

First step:

Collect data in form of a table.
Time of Fall (seconds) Distance of Fall (meters)
1 5
2 20
3 45
4 80
5 125
Here the time is measured in seconds and the distance in meters.

Those quantities are fundamental units The fundamental units are (SI-Units)

time second s
length meter m
mass kilo gram kg
temperature Kelvin K

Second step:

These data could also be presented in the form of a graph, in which distance is placed against time:

Third step:

After preparing tables and graphs of their data. Scientists would notice that the larger something falls, the further it travels. If an object falls for twice as long as another, it will travel four times as far; if it falls three times larger, it will travel nine times as far, and so on. This statement can be summarized in these ways:

In words:

The distance traveled is proportional to the square of the time of travel.

In the form of an equation:

Distance = constant x (time)2

In symbolic form:

d = c x t2

(The constant c has to be determined from the measurement)

Mathematics is a concise language that allows scientists to communicate their results to make very precise predictions, but anything that can be said in an equation can also be said (allied in a less concise way) in a plain English sentence. When encountering equations, you should always ask `What English sentence does this equation represent?'. This nature will keep the mathematics from obscuring the simple ideas that lie behind most equations.

Ch. Elster
Aug 26 14:27:03 EDT 2019