Motion is a change of the position of an object from one instant to the next.... so to start the task of describing motion, we ask: what is position? and what physical quantities do we use to describe a change in position?

Position is relative

• The first thing to know about position, is that it is a relative quantity -- meaning that it has to be defined relative to some frame of reference.

Later on (Physics 3) we will learn how to deal with moving reference frames. It was the long standing theory that there existed a universallystationary frame, a frame which was stationary to everything inthe universe, until the famous Michelson-Morley experiment provedotherwise. A few years later, Albert Einstein formulated the theory of special relativity which provided the framework to understand the universe with no absolutely stationary frames.

• Watch (one or both of these) videos, as you contemplate the idea of relative position, and relative motion.

The Position Vector

Step 1: Set up a Reference Frame and a Coordinate System.

In setting up a coordinate system, we must choose the positive, and therefore negative, direction of our chosen axes. We will choose our axis to be the -axis, and select the direction that the car is facing as the positive direction, i.e. the direction represented by unit vector . Note that you may choose whatever arbitrary label you'd like for the x-axis, but this is the most common choice and the notation we will use in these modules.

 

Step 2: Approximate the object in question as a point.

This is done by choosing a point on the object arbitrarily. The particular choice of the point is irrelevant, so long as it is consistent during problem solving. For convenience, we will select the center of mass of the car as our point in question.

Once we have established our reference frame and our point particle in question, we can describe the position of the car by using a Position Vector.

Suppose the car is at position , which is  meters in the  direction from the zero point of our system. Its position vector ​ is given by:

The + sign and the unit vector indicate that the car is in the positive direction relative to the zero point. Note that the SI unit for the measurement of position is given by the meter ().

If a second car is at position B, which is away from the zero point in the opposite direction as position , then the position vector of the second car is given by:

We will discuss multi-dimensional vectors later in this module,
 
but for now, just note that  is the unit vector along the x-axis, ​ is the unit vector along the y-axis, and  is the unit vector along the z-axis.

---

The Displacement Vector

Suppose the car is initially in position A given by the position vector:

At some later time, the car has moved to position B given by the position vector:

We describe this motion mathematically by a Displacement Vector defined by:

The Greek letter capital Delta  is often used to denote a change in a quantity, in this case, a change in position.

There are a few things worth stating explicitly:

• The displacement vector  is defined as the final position minus the initial position:

• The displacement of an object during any segment of motion can be thought of as the position of the object at the end of the segment relative to its position at the beginning of the segment.

•  Displacement is a vector, which means it represents both a magnitude ($2$ meters) and a direction (). This indicates that the car is traveling in the opposite direction from which it is facing, in a typical case, backing up. The reverse orientation of the displacement vector (initial position minus final position) would indicate the car moving forward, hence the importance of the order.

• The SI unit for the magnitude of displacement is meters, which is consistent with our previous units.

---

The Distance Scalar

Another commonly used quantity used to describe motion is the Distance Scalar, or distance traveled. There are two fundamental differences between displacement and position:

1. Distance traveled is a scalar, so it does not contain information about the direction of motion. In the previous example, the distance traveled can be , but we do not know if the overall motion was forward or backward.

2. Distance is defined by the length of the path of an object, whereas displacement is defined by the initial and final positions of an object.

To illustrate this difference, consider if the car in the previous example first moved forwards , then backed up . The displacement vector is still as it only depends on the initial and final positions of the car. However, the distance traveled by the car would then be .

---

In summary, distance is a path dependent scalar quantity, whereas displacement is a path independent vector quantity.

---

Watch this summary: