# Laboratory

## Just kidding!

There is some math in this exercise. It is not as difficult as it might appear. We need to learn to use arithmetic in order to see relationships. This is a slightly more sophisticated use of numbers than counting, adding, or subtracting.

We want to explore the concept of RATIO, PROPORTION and the utility of a graph to visualize relationship of different kinds.

A ratio is a fraction, the process of division of two numbers expressed as "y" "divided" by "x". Where y and x represent any two numbers. We write this as as y/x and read it "y over x", or "y divided by x"

Ratios are common in everyday activities. In everyday usage we call them "rates". Ratio, rate, ration, rational, rationale are all related words. Do you see the connection?

We might want to know how many MILES PER HOUR our car is moving. We might be concerned about MILES PER GALLON as a measure of our car's fuel efficiency. We might want to know the PRICE PER POUND of ahu sashimi before we decide how much to buy. Or we might want to know the RATE of interest our bank is paying or charging.

All of these are ratios. They all represent a CHANGE of one kind or another. Specifically they all designate how fast one thing is changing with respect to another.

CHANGE is one of our topics of interest in the physical sciences. It is an interesting word (look it up), and likewise an interesting concept.

Early mathematicians, Aristotle among them, taught that mathematics was of no use in describing change. Today we disagree. In fact the mathematics of change is our primary mathematical tool in physical science.

If something is different now from the way it was before we say it has "changed". Qualitative changes are easy to perceive visually because our brains are "wired" to recognize movement. You may recall that Aristotle did not distinguish between motion and change, and in fact considered the movements of the planets not as change but as constancy.

We are usually aware of changes in speed because we have speedometers to measure it, and also because changes in speed are felt as forces. We will study forces in a later exercise. If we bought gasoline and sashimi often enough we would be aware of CHANGES in these two numbers too.

It is, for example, much easier to see something which moves against a complex background than to see the same something when standing still. That's why camouflage works, and that's why we find it easier to see changes in ratios when we plot data as a graph.

Quantitatively, we can use numbers to measure change as long as we are comparing two things which can actually be measured. Physical quantities are those which can be measured. A ratio is one way of comparing two things.

A ratio can also be viewed as a scale model. For example a model ship at a scale of 1 to 10 means that everything on the model is exactly ten times smaller than on the real ship. The ratio is 1/10. One foot on the times smaller than on the real ship. The model ship is exactly one - tenth the size of the real ship.

There is a CONSTANT RATIO of length between the real ship and the model ship.

The same could be said for an object and a photograph of it, and between the photograph and the negative. Or between a photograph and an enlargement. All demonstrate the concept of CONSTANT RATIO.

Wait. You may be thinking, "Earlier he was writing about change and now all of sudden there is this stuff about CONSTANT RATIOS. So which is it, constant or change?"

OK it may seem like I strayed, but the concepts of change and constant are related because change can occur at a constant rate. Change may also occur at a changing rate, and even that can be represented and recognized graphically.

Certain types of change can be represented as a constant ratio. This will show up as a straight line on a graph if the graph is plotted the correct way. When the ratio of two quantities is constant we say there is a linear relationship between them. It also allows a way to easily see whether or not such a relationsip exists. This is a powerful technique for experiments because we can use a graph to verify or negate a suspected relationship.

If a ratio is constant, it means that the two quantities change by the same factor so their ratio amounts to the same number. A CONSTANT RATIO tells us that two things change at the same rate or are related to one another by the same scale factor.

It is up to us to interpret the meaning and usefulness of the information.

When the ratio is not constant the data will not plot as a straight line and we may need to use other tactrics to change it into a straight line. You will do this in the final part of lab 3.

If a ratio is changing, we find it useful to concern ourselves with how fast it is changing.

Have fun, and stay tuned.