Position, Distance, and Displacement

1. Position

Definition

Position describes where an object is located relative to a reference point (origin).

In one dimension:

$$x(t)$$

In three dimensions:

\(\vec{r}(t)=x(t)\hat{i}+y(t)\hat{j}+z(t)\hat{k}\)

• \(\vec{r}\) is the position vector
• It points from the origin to the object

2. Distance

Definition

Distance is the total length of the path traveled by an object.

Characteristics:

• scalar quantity (no direction)
• always nonnegative
• depends on the actual path taken

Example:

If an object moves
3 m forward, then 2 m backward

Distance:

3m+2m=5 m

3. Displacement

Definition

Displacement is the change in position.

Δr⃗=r⃗f−r⃗i\Delta \vec{r} = \vec{r}_f – \vec{r}_iΔr=rf​−ri​

where:

• r⃗i\vec{r}_iri​ = initial position
• r⃗f\vec{r}_frf​ = final position

Displacement is a vector, so it has both magnitude and direction.

4. Displacement in One Dimension

Δx=xf−xi\Delta x = x_f – x_iΔx=xf​−xi​

Example:

xi=2 m,xf=10 mx_i = 2\,\text{m}, \quad x_f = 10\,\text{m}xi​=2m,xf​=10m $$\Delta x=8\text{m}$$

5. Distance vs Displacement

Example:

Move 5 m east, then 5 m west:

Distance:

$$10\text{m}$$

Displacement:

$$0$$

6. Displacement in Two Dimensions

If an object moves from:

(xi,yi)→(xf,yf)(x_i, y_i) \rightarrow (x_f, y_f)(xi​,yi​)→(xf​,yf​)

Then:

Δr⃗=(xf−xi,  yf−yi)\Delta \vec{r} = (x_f – x_i,\; y_f – y_i)Δr=(xf​−xi​,yf​−yi​)

Magnitude:

∣Δr⃗∣=(Δx)2+(Δy)2|\Delta \vec{r}| = \sqrt{(\Delta x)^2 + (\Delta y)^2}∣Δr∣=(Δx)2+(Δy)2​

This follows:

7. Physical Meaning

• Position → where the object is
• Distance → how much ground it covered
• Displacement → how far and in what direction it moved overall

These distinctions are essential because:

• velocity is based on displacement
• speed is based on distance

Summary

Position:

r⃗(t)\vec{r}(t)r(t)

Distance:

• total path length (scalar)

Displacement:

Δr⃗=r⃗f−r⃗i\Delta \vec{r} = \vec{r}_f – \vec{r}_iΔr=rf​−ri​

• vector quantity
• includes direction

These concepts form the foundation of all motion analysis in kinematics.