Displacement

You walk from your house to a shop — winding through lanes, doubling back once. Your distance is every step you took. Your displacement is a straight arrow drawn from your front door to the shop door. Only the endpoints matter.

This is the first place physics parts ways with everyday language. "How far did you go?" (distance) and "how far are you from where you started?" (displacement) are two different questions with two different answers.

If a particle moves from position $\vec{r_i}$ to position $\vec{r_f}$:

$\vec{s} = \vec{r_f} - \vec{r_i}$

Displacement is a vector — it has magnitude and direction. Its SI unit is the metre (m).

Key properties:

• Displacement can be zero even when distance is non-zero (return to start)
• $|\vec{s}| \leq$ distance (equality only when path is a straight line, no backtracking)
• Displacement depends only on initial and final positions

No derivation in the classical sense — displacement is a definition. It is the vector difference of two position vectors measured from a common origin.

The origin choice does not affect displacement: if you shift the origin by $\vec{a}$, both $\vec{r_f}$ and $\vec{r_i}$ shift by $\vec{a}$, and the difference $\vec{r_f} - \vec{r_i}$ is unchanged.

1. Average velocity — defined only using displacement, not distance: $\vec{v}_{avg} = \frac{\vec{s}}{\Delta t}$ Speed uses distance; velocity uses displacement. This distinction matters everywhere in kinematics.

2. Projectile motion — the displacement vector at any instant $t$ is: $\vec{s}(t) = (u\cos\theta)\,t\,\hat{i} + \left[(u\sin\theta)\,t - \tfrac{1}{2}gt^2\right]\hat{j}$

3. Circular motion — a particle completing one full revolution has zero displacement but non-zero distance ($2\pi r$). This is the clearest example of the distinction.

"Displacement is always smaller than distance." It is always less than or equal to distance. They are equal when the path is a straight line with no reversal. Students lose marks by writing "smaller than" instead of "less than or equal to."

"Displacement cannot be negative." It can. A negative component of displacement simply means the particle moved in the negative direction along that axis. Sign carries physical meaning.

"Displacement and distance have the same unit so they are the same thing." Same unit (metre), completely different quantities. One is scalar, one is vector. One depends on path, one does not.