2.1 Displacement

Understanding Displacement and Position

Position and displacement are the starting points for describing motion in physics. Position tells you where an object is relative to some chosen reference point, while displacement tells you how that position changed. Getting these concepts straight now makes everything else in kinematics easier to follow.

Position and Displacement Concepts

Position specifies an object's location in space relative to a reference point (called the origin). You describe it using coordinates, like $(x, y)$ in 2D or $(x, y, z)$ in 3D, and represent it with a position vector ($\vec{r}$).

Displacement is the change in an object's position. It only cares about where you started and where you ended up, not the path you took to get there.

• Represented by $\Delta\vec{r} = \vec{r}_f - \vec{r}_i$
• It's a vector, so it has both magnitude (how far) and direction (which way)
• If you walk from your dorm to the library and back, your displacement is zero even though you clearly moved

Distance vs. Displacement

These two get confused constantly, but they measure different things.

Distance is a scalar (just a number). It measures the total path length you traveled, and it's always positive or zero. Think of it as what your odometer reads.

Displacement is a vector. It measures the straight-line change from your starting position to your ending position, and it can be positive, negative, or zero depending on direction.

Why does this matter? Distance is useful when you care about total effort or energy spent. Displacement is what you need when analyzing the net result of motion, which is what most kinematics problems ask about.

Calculating Displacement

To find displacement, subtract the initial position from the final position:

• 1D: $\Delta x = x_f - x_i$
• 2D: $\Delta\vec{r} = (x_f - x_i)\hat{i} + (y_f - y_i)\hat{j}$
• 3D: $\Delta\vec{r} = (x_f - x_i)\hat{i} + (y_f - y_i)\hat{j} + (z_f - z_i)\hat{k}$

For this course, you'll mostly work in 1D. If an object starts at $x_i = 2 \text{ m}$ and ends at $x_f = 7 \text{ m}$, the displacement is $\Delta x = 7 - 2 = 5 \text{ m}$ in the positive direction. If it moved the other way (from 7 m to 2 m), the displacement would be $-5 \text{ m}$, and that negative sign tells you the direction.

The path taken between the two positions doesn't affect the result. Displacement depends only on the start and end points.

When Distance Equals Displacement

Distance and displacement have the same magnitude only when an object moves in a straight line without changing direction. The moment the object curves, zigzags, or reverses, distance will be greater than the magnitude of displacement.

• A car driving one full lap around a circular track covers a large distance but has zero displacement (it returns to its starting point).
• A ball rolling 3 m to the right in a straight line has both a distance and a displacement magnitude of 3 m.

Motion and Kinematics

A few foundational terms to keep straight:

• Motion is a change in position over time.
• Kinematics is the branch of physics that describes motion (position, displacement, velocity, acceleration) without worrying about why the motion happens. Forces come later.
• Trajectory is the path a moving object traces through space.
• Reference frame is the coordinate system you choose to describe position and motion. Your displacement values depend on which reference frame you pick, so always be clear about your origin and axes.