3.2 Displacement

Introduction

Displacement is a fundamental concept in kinematics that describes an object's change in position. It is defined as the shortest distance between the initial and final points of an object's motion, combined with the direction of that change. As a vector quantity, displacement provides a more complete picture of motion than the scalar quantity of distance.

Definition and Vector Nature

Displacement ($\vec{d}$) is a vector quantity representing the overall change in an object's position. It has two key components:

• Magnitude: The length of the straight line connecting the starting point to the ending point.
• Direction: The direction of the straight line from the start point to the end point.

The SI unit for displacement is the meter (m) and its dimensions are $[L]$.

Displacement versus Distance

This is one of the most important distinctions in kinematics.

• Distance is a scalar quantity that measures the total path length covered during a journey. It only has magnitude.
• Displacement is a vector quantity that measures the net change in position. It is independent of the path taken.

Example: A person walks 4 meters East and then 3 meters North.

• Distance traveled: $4\text{m}+3\text{m}=7\text{meters}$
• Displacement: The magnitude is $\sqrt{4^2+3^2}=5\text{meters}$. The direction is Northeast. The displacement is 5 meters, Northeast.

Scalar And Vector Quantities→

Representation using Position Vectors

Displacement can be precisely calculated as the difference between an object's final and initial position vectors. A position vector is a vector that represents the location of a point relative to an origin.

• Let ${\vec{r}}_1$ be the initial position vector (from the origin to the start point).
• Let ${\vec{r}}_2$ be the final position vector (from the origin to the end point).

The displacement vector $\Delta \vec{r}$ (or $\vec{d}$) is given by:

$\Delta \vec{r}={\vec{r}}_2-{\vec{r}}_1$

This equation finds the vector that connects the tip of the initial position vector to the tip of the final position vector.

Zero and Negative Displacement

• Zero Displacement: An object can have zero displacement even if it has traveled a significant distance. This occurs when the object returns to its starting point, such as completing one lap around a circular track. Here, the initial and final positions are the same, so $\Delta \vec{r}=0$.

• Negative Displacement: The sign of displacement indicates direction relative to a chosen coordinate system. Negative displacement means the object moved in the negative direction (e.g., left, down, or south) from its starting point.

Key Questions

Q: Can the magnitude of displacement be greater than the distance traveled?

A: No. The magnitude of displacement is the shortest distance between two points, so it can only be less than or, in the case of straight-line motion in one direction, equal to the distance traveled.

Q: How is displacement represented graphically?

A: Displacement is represented by an arrow drawn from the object's initial position to its final position. The length of the arrow is proportional to the magnitude of the displacement, and the arrowhead indicates the direction.

Summary

• Displacement is a vector quantity that describes the change in an object's position.
• It is the shortest path from the initial to the final point, including direction.
• It is calculated as the final position vector minus the initial position vector: $\Delta \vec{r}={\vec{r}}_2-{\vec{r}}_1$.
• Displacement is path-independent, unlike distance.

Understanding displacement is crucial as it forms the basis for defining other key vector quantities in physics, such as velocity and acceleration.