Understanding Scalars and Vectors in Motion

Position

Position is a vector quantity that identifies the location of an object relative to a chosen reference point, often called the origin. In a one-dimensional coordinate system, we represent position with the variable $x$. If an object is at $x = 5\text{ m}$, it is five meters to the right of the origin.

Displacement

While position tells us where an object is at a specific moment, displacement tells us how much the position has changed over a period of time. Displacement is a vector quantity representing the straight-line change in position from a starting point to an ending point.

The formula for displacement ($\Delta x$) is:

$$\Delta x = x_{final} - x_{initial}$$

It is crucial to note that displacement only cares about the "start" and the "end." It does not care about the path taken in between.

Describing Path Length: Distance

Distance is a scalar quantity that represents the total length of the actual path traveled by an object. Unlike displacement, distance is always cumulative and cannot be negative.

Distance vs. Displacement: A Comparison

Consider a runner on a circular track with a circumference of 400 meters. If the runner completes exactly one full lap:

1. Distance: The runner has traveled 400 meters.
2. Displacement: Since the runner ended exactly where they started, their change in position is zero ($\Delta x = 0$).

Describing Rate of Motion: Speed and Velocity

Once we know how far an object has moved and how long it took, we can describe its rate of motion.

Speed (The Scalar Rate)

Speed is a scalar quantity that describes how fast an object is moving. It is defined as the rate at which distance is covered over time. Because it is a scalar, it does not include direction.

The formula for average speed is:

$$\text{average speed} = \frac{\text{total distance}}{\text{time interval}}$$

Velocity (The Vector Rate)

Velocity is a vector quantity that describes the rate at which an object changes its position. It is essentially "speed with a direction." Velocity is defined as the displacement divided by the time interval.

The formula for average velocity ($v_{avg}$) is:

$$v_{avg} = \frac{\Delta x}{\Delta t} = \frac{x_{final} - x_{initial}}{t_{final} - t_{initial}}$$

Because velocity is based on displacement, it is possible for an object to have a high speed but an average velocity of zero (as seen in the circular track example above).

Worked Example: The Commuter's Journey

Let's apply these concepts to a real-world scenario to ensure the distinctions are clear.

Scenario: A student walks 30 meters East from their house to a coffee shop, realizes they forgot their wallet, and walks 10 meters West back toward their house. The entire trip takes 100 seconds.

1. Calculate the total distance: The student walked 30m and then 10m.

$$\text{Distance} = 30\text{ m} + 10\text{ m} = 40\text{ m}$$

2. Calculate the displacement: Taking East as the positive direction and West as negative:

$$\Delta x = (+30\text{ m}) + (-10\text{ m}) = +20\text{ m (or 20 m East)}$$

3. Calculate the average speed:

$$\text{Speed} = \frac{40\text{ m}}{100\text{ s}} = 0.4\text{ m/s}$$

4. Calculate the average velocity:

$$v_{avg} = \frac{20\text{ m}}{100\text{ s}} = 0.2\text{ m/s East}$$

As demonstrated, even though the student was moving throughout the entire 100 seconds, their velocity is much lower than their speed because their net change in position was smaller than the total path they walked.