Defining Motion: Position, Displacement, Distance, and Path Length

1. The Language of Motion: Frames of Reference and Point Particles

Before we can describe motion, we must agree on two simplifying abstractions.

First, in kinematics, we often model real-world objects (cars, planets, electrons) as point particles. This is an idealization where we consider an object's size and shape to be negligible for the problem at hand, allowing us to describe its location as a single point in space. This is a powerful simplification that lets us focus purely on the object's translational motion.

Second, any description of motion is meaningless without a point of view. In physics, this "point of view" is formalized as a frame of reference. A frame of reference consists of:

• An origin: A fixed point O from which we make our measurements.
• A coordinate system: A set of axes (e.g., Cartesian x, y, z axes) that define directions in space.
• Clocks: To measure time.

All motion is relative. The numbers you use to describe a particle's location depend entirely on the frame of reference you choose.

2. Position: Where is the particle?

With a frame of reference established, we can now rigorously define position.

The position of a particle is its location in space relative to the origin of the coordinate system. We represent this mathematically using the position vector, denoted by $\vec{r}$.

The position vector $\vec{r}$ is a vector drawn from the origin of the coordinate system to the current location of the particle.

To understand this formally, please read the following introductory sections.

As you've read, in a 3D Cartesian system, the position vector can be written in terms of its components and the standard unit vectors $\hat{i},\hat{j},\hat{k}$:

Here, $(x,y,z)$ are the coordinates of the particle. The magnitude of the position vector, $|\vec{r}|$ or simply $r$, is the particle's straight-line distance from the origin:

Crucially, position is a vector. It has both magnitude (how far?) and direction (where?).

3. Displacement: The Change in Position

Motion is fundamentally about change in position. This change is captured by a new vector: the displacement vector.

The displacement of a particle is the vector representing the change in its position.

Let's say a particle is at an initial position ${\vec{r}}_i$ at time $t_i$ and moves to a final position ${\vec{r}}_f$ at time $t_f$. The displacement vector, $\Delta \vec{r}$, is defined as the vector difference:

This vector points from the initial position to the final position. The diagram below provides an excellent visualization of this vector subtraction.

The components of the displacement vector are simply the differences in the corresponding components of the position vectors:
If ${\vec{r}}_i=x_i\hat{i}+y_i\hat{j}+z_i\hat{k}$ and ${\vec{r}}_f=x_f\hat{i}+y_f\hat{j}+z_f\hat{k}$, then

4. Distance vs. Path Length: The Heart of the Distinction

Now we arrive at two concepts that are often confused: distance and path length. Your preference for rigor will be particularly helpful here.

• Path Length ($s$): The path length is the total length of the actual path, or trajectory, that the particle travels. Imagine an odometer in a car; it measures path length. It is a scalar quantity and is always non-negative.
• Distance ($d$): The distance is defined as the magnitude of the displacement vector, $d=|\Delta \vec{r}|$. It represents the length of the straight line connecting the initial and final points. It is also a scalar.

The shortest path between two points is a straight line. Since distance is the length of this straight line, and path length is the length of the actual, possibly meandering, path, we can state a fundamental inequality:

Distance $\le$ Path Length

Let's prove this with the mathematical rigor you appreciate.

The Proof using the Triangle Inequality

Consider a particle moving along a curved trajectory from point A to B. We can approximate this curve as a sequence of $N$ infinitesimally small, straight-line displacement vectors: $\text{d}{\vec{r}}_1,\text{d}{\vec{r}}_2,\dots ,\text{d}{\vec{r}}_N$.

1. The total displacement vector $\Delta \vec{r}$ is the vector sum of these infinitesimal displacements:

   $$\Delta \vec{r}=\sum _{k=1}^N\text{d}{\vec{r}}_k$$

2. The total path length $s$ is the scalar sum of the magnitudes of these infinitesimal displacements:

   $$s=\sum _{k=1}^N|\text{d}{\vec{r}}_k|$$

3. By applying the triangle inequality repeatedly (a process known as mathematical induction), we can extend it from two vectors to $N$ vectors:

   $$\left|\sum _{k=1}^N\text{d}{\vec{r}}_k\right|\le \sum _{k=1}^N|\text{d}{\vec{r}}_k|$$

4. Substituting our definitions from steps 1 and 2, we arrive at the proof:

   $$|\Delta \vec{r}|\le s$$

   Or, in other words, distance $\le$ path length.

The equality $d=s$ holds if and only if the particle moves along a straight line and does not reverse its direction. In this specific case, all the infinitesimal vectors $\text{d}{\vec{r}}_k$ point in the same direction.

A Concrete Example

Imagine a particle moving from point A to point B along a semicircular path of radius $R$, as shown below. Let's place the origin at the center of the circle.

A particle travels from A(-R, 0) to B(R, 0) along a semicircular path.

• Initial Position: ${\vec{r}}_A=-R\hat{i}$
• Final Position: ${\vec{r}}_B=+R\hat{i}$
• Displacement: $\Delta \vec{r}={\vec{r}}_B-{\vec{r}}_A=(R\hat{i})-(-R\hat{i})=2R\hat{i}$
• Distance: $d=|\Delta \vec{r}|=|2R\hat{i}|=2R$
• Path Length: $s$ = length of the semicircle = $\frac{1}{2}(2\pi R)=\pi R$

As you can see, the distance ($2R$) is indeed less than the path length ($\pi R$), since $\pi \approx 3.14>2$.

Conclusion

This lesson established the fundamental vocabulary of kinematics with the rigor you requested. We have built these concepts from the ground up, starting with the need for a frame of reference.

Here are the key takeaways:

Most importantly, we rigorously proved the relationship distance $\le$ path length, with equality holding only for unidirectional, straight-line motion.

In our next lesson, we will continue building on this vector foundation. We'll explore how to apply vector algebra to decompose motion in arbitrary coordinate systems. This will involve resolving vectors into components, which is the critical step that bridges pure mathematical definitions with the practical analysis of motion in two and three dimensions.