Distance and Displacement

Understanding Distance and Displacement

In everyday language, we often use "distance" to describe how far apart things are. However, in Physics, we must distinguish between the actual path taken and the net change in position.

1. The Definitions

Distance ($s$)

A scalar quantity that measures the total length of the actual path traveled between two locations. It only tells you "how much ground was covered."

Displacement ($\mathbf{s}$ or $\Delta \mathbf{s}$)

A vector quantity that measures the interval between two locations along the shortest path connecting them. It tells you "how far out of place" an object is and in what direction.

The Rule of Magnitude

The magnitude of displacement is always less than or equal to the distance traveled. They are only equal when motion occurs in a straight line without changing direction.

2. Comparison: How Far is it?

We often state distances in terms of time (e.g., "It's a 90-minute drive"), but in Physics, distance must be measured in units of length.

Scenario	Possible Answer (Time)	Standard Physical Distance
Earth to Sun	8.3 light-minutes	$1.5 \times 10^{11} \text{ m}$ (1 AU)
Space Station Orbit	90 minutes	$40,000,000 \text{ m}$
Central Park to Battery Park	90 minutes on foot	$10,000 \text{ m}$

3. Practical Example: The Commuter’s Path

Imagine a journey from a starting point to a store across a river:

Travel 8.2 km North along a river.
Cross a bridge 1.8 km Wide.
Travel 4.5 km South to reach the destination.

Total Distance

$8.2 + 1.8 + 4.5 = \mathbf{14.5 \text{ km}}$.

Displacement

Since you went 8.2 km North and 4.5 km South, your net vertical change is only 3.7 km North. Combined with the 1.8 km bridge width, your straight-line displacement is much smaller than the 14.5 km you traveled.

4. Notation and Symbols

While many use $d$ or $x$, standard physics often uses the Latin root spatium (meaning "space" or "distance").

$s$ = Distance (Scalar, italicized)
$\mathbf{s}$ = Displacement (Vector, bolded)
$\Delta s$ = The "Change" in position.

5. Standard Units of Measurement

The SI unit for both is the meter [m]. However, different scales require different units:

Nautical Mile: $1,852 \text{ m}$. Originally defined as $1/60$ of a degree of the Earth's circumference. Used in aviation and shipping.
Earth Radius ($R_E$): $6.4 \times 10^6 \text{ m}$. Used to describe satellite orbits or planetary sizes.
Astronomical Unit (AU): $1.5 \times 10^{11} \text{ m}$. The average distance from Earth to the Sun.
Light Year (ly): $9.5 \times 10^{15} \text{ m}$. The distance light travels in a vacuum in one year.

6. The Nature of Space (Symmetry)

Physics assumes that space has specific properties that make our measurements reliable:

Homogeneous (Translation Symmetry): It doesn't matter where you put your "zero" point (origin). The distance between two points remains the same.
Isotropic (Rotation Symmetry): It doesn't matter which way your axes point (North/South or East/West). The length of a vector is independent of its orientation.
Chirality (Reflection Symmetry): Most basic quantities like distance don't change if you look at them in a mirror. (Note: This is not true for "pseudovectors" like magnetic fields!)