# Reference Frames, Distance, Displacement

*Reference
Frames*

A reference frame allows us to set a context for examining a problem, and
all measurements in Physics need to be made with respect to some frame of reference. For example, if you are seated on the Downeaster heading

towards Boston, in your frame of reference (relative to the train), you are not moving. But to an observer standing by the side of the tracks, you are moving at a speed of 40 mph. Now, get up and
start walking towards the front of the train at 5 miles per hour. To the observer by the tracks, you are now moving at 45 miles per hour.

An inertial reference frame is one that is not experiencing acceleration. A
non-inertial reference frame, therefore, is one with acceleration present. Consider a leisurely drive down Route 17 towards Kents Hill. You've just stopped at a Dunkin Donuts and your coffee is
sitting comfortably on the dashboard as you drive. With no acceleration (not a likely scenario as the road is hilly and there is always acceleration in some direction), it is impossible to
measure the speed of your car, or the motion of your car from inside. A ball tossed straight up by the passenger next to you falls straight down. Your coffee stays on the dashboard.
Enter a moose (after all, this is Maine.) You immediately apply your brakes to
avoid hitting this magnificent creature that has parked itself in the middle of the road, changing your frame of reference to a non-inertial frame and projecting your coffee across the windshield of
your car. You are subject to the laws of motion, and acceleration now plays a big

role in your life. Einstein based his postulates for his special
theory of relativity on this. His First Postulate, the Relativity Principle, states that the laws of physics have the same form in all inertial reference frames. For the majority of our work in Physics, the Earth will be our inertial reference frame.
Although, in the context of the solar system and the universe, Earth is rotating and hurtling through the galaxy, we will consider an object at rest *with respect to the surface of the
Earth*to truly be at rest.

*Distance
and Displacement*

When we ask how far away something is, we are talking about distance. In
physics, we will talk about 2 distinct terms, *distance* and *displacement*. In order to do that, however, we need to define two terms: *vector* and *scalar*

quantities.

***Vector** quantities have
magnitude and direction.

***Scalar** quantities have
only magnitude.

For example, when I say Augusta is 15 miles away, I am referring to a scalar quantity. If I say Augusta is 15 miles east of here, then I am referring to a vector quantity.

Distance and Displacement

There is
an important difference between **distance** and **displacement**. Distance is a scalar quantity, and represents the total distance traveled. Displacement is a vector
quantity and represents the straight-line distance from the origin. Physically, we say *displacement is the change in position of an object.* These quantities are measured using an axis (a
reference frame) with set distances in any direction from the origin. Displacement values can be positive or negative, depending on the direction traveled along the axis from the origin. Often,
we

will choose the reference frame and origin as the most convenient values for our problem.

Displacement = (final position) – (starting position)

ΔX = X_{2} – X_{1}

Let's take this opportunity to define some symbols. In our classes, we will use the symbol X to denote position.

- The initial quantity of any value will be denoted using either
the subscript "i" or the subscript "0", as in X
_{i}or X_{0}. (This is done to be consistent with the textbooks in use - personally, I will generally default to the 0 subscript. Sorry.) - The final value will generally just be given by the Symbol without a subscript. For example, final position is X.
- A "change in value" (Δ) will always be the final minus
the initial. This gives us a chance to be consistent. For example,

Displacement, defined as ΔX, is equal to final position minus initial position. (ΔX = X– X_{0)}) - When we are talking about distance (the scalar), we may use the symbol "d".
- Finally, many texts use the symbol "S" as displacement. Go figure.

Example: A person walks 20 meters east and then turns around and walks 5 meters west.

Distance = 25 meters (total distance traveled)

Displacement = 20 meters + (-5) meters = 15 meters (straight-line distance from the origin with east being in the positive direction.)

Another example: A person walks 3 miles north and then walks 4 miles east.
The total distance walked was 7 miles. But the displacement from the origin was 5 miles to the (roughly) northeast. (We will use trig in another

lesson to determine the actual direction traveled.)

A final example: A bored physics student (as if there could be such a
creature) is sitting at his lab table (measuring 1.2 m by .7 m). He starts with his book in one corner and moves it around the perimeter of the

table until it is back where it started. The distance the book traveled was .7m + 1.2m + .7m + 1.2m = 3.8m. But the displacement of the book is 0 meters - it is right back where it started with no
final displacement from the origin.

It is important to note that when we talk about distance, we will not use a
direction. When we discuss displacement, we will note a direction. Often, it may be as simple as assigning a direction (up or down,

left or right) to positive and negative values. Graphing is an
important tool for helping us visualize the

motion of an object. We will look at a series of graphs in our studies, but let's start with a basic displacement versus time graph. In this type of graph, the
vertical (Y) axis represents our displacement from the starting point (origin) and the horizontal (X) axis represents the time elapsed from the start of the event. Consider the following:

A student starts at the origin and walks 6 meters away taking a time of 3
seconds to do so. The student then stands still for 2 seconds, and walks back towards the origin a distance of 3 meters in the next 5 seconds. What does the displacement vs. time graph look like for
this event?

We will take many opportunities to work on these graphs in class including some lab work!

Often it is helpful to see what someone else has to say on the
topic. Try http://www.physicsclassroom.com/Class/1DKin/U1L1c.html

for more information on distance and displacement. Also check out http://www.physicsclassroom.com/Class/1DKin/U1L3a.html

for more information on position vs. time curves. Go ahead - you will need to know this stuff!