You probably have most of this stuff at home anyway

1.2 Frames of Reference, Position, and Displacement

Every lesson will begin with a few key concepts

Key concepts: 

  • A frame of reference defines a zero point for our measurements and tells us which direction will be positive.

  • Within a frame of reference, an object’s position can be expressed by a coordinate point.

  • When an object’s position changes, the object is “displaced.” Displacement is the change of position.

Every lesson will include textbook sections

Textbook section: 2.1 - 2.2

A frame of reference is basically a coordinate system that we put on nature in order to figure out where things are.   Here's an example:

This site is well worth a visit: https://openpress.usask.ca/geolmanual/chapter/overview-of-topographic-maps/

Notice the numbers on the top and right sides of the map?  Those are coordinates. They are in latitude, longitude format, which is not quite the same as a simple x-y plane. However, over small distances, it’s close enough. For those who want to know: latitude and longitude are actually measured in terms of angles using the center of the earth as the origin of coordinates. The earth is approximately a sphere, so when we translate those lines to a sheet of paper to make a map it’s just an approximation. As long as we don’t try to map huge areas, the approximation is good enough for our purposes.

Now, we don't have LITERAL lines drawn across the earth... Most of the time, anyway.

Thanks: http://www.carnegieknowledgenetwork.org/briefs/value-added/level-playing-field/

A frame of reference has two jobs: It tells us where we’re going to put the zero point for our measurements, and it tells us which way (or ways, if we’re in more than one dimension) will be positive. In its simplest form, a frame of reference is just a number line in one dimension or a coordinate plane in two dimensions.

Position:

Once we have a frame of reference, position is the coordinate within that frame of reference. Position is, fundamentally, the answer to two questions:

  • How far are you from zero?

  • In which direction?

Because position is a distance, it is measured using distance units. In physics, we usually measure position in meters, but for small distances we might use centimeters. Position will be written as one signed number per dimension, so you might see something like: x = -17 m. That means the object is 17 meters from the zero point, and it is on the negative side.

Displacement:

Displacement is the change of position, regardless of the path taken. Thinking of the football field shown above, the object of the game — if you’re on offense — is to move the ball 10 yards from the line of scrimmage. HOW your team moves the ball is not important. The path that the ball takes may be quite a lot longer than 10 yards. “Yardage gain” (or loss…) is an example of displacement.

In math, displacement is easy:

Notice a couple of things: First of all, we use a delta (the triangle shaped thing) to indicate a change in something. This notation is pretty universal; anytime you see delta-something, it means the change in that variable. Second, the unit for displacement is the same as the unit for position. It will usually be meters. Third, notice that we are using subscripts to indicate x-final and x-initial. Different textbooks use different lettering systems. Sometimes you will see “f” for final and “I” for initial, and sometimes you will see something else. At the college level, you will almost always see the following notation:

In this style of notation, a zero subscript indicates the initial value, and a variable without a subscript is the current, or final value. Think of the subscript as indicting time. Zero time is the initial condition.

Of course, an object can move in more than one dimension, so we might also have a displacement in the y direction:

You can probably see how this would extend into three dimensions with a delta-z variable. You will see this basic idea extended all over the place, well beyond just position, so make sure that you understand it now!

Distance:

Remember that displacement is like yardage gain or loss in (American) football. If you have seen the game being played, you might also know that a ball can go quite a long ways and end up right back on the starting point. (The line of scrimmage.) The displacement of that ball would be zero. However, the distance that it has traveled is not zero. Your intuition for distance — that it is how far you have gone, including all the twists and turns — is correct. Distance is always a positive number.

Each unit will end with a to-do list.

To Do: