Position, distance and displacement Knowledge-Based Study

Study notes

1. Reference Frame and Position

   To describe motion accurately, we must first specify where an object is located relative to a chosen landmark called the reference point or origin ($x = 0$).

   Position is a vector quantity that defines the location of an object at a particular instant in time relative to the chosen origin. In one-dimensional motion along a straight line:

   • Position is denoted by the symbol $x$.
   • It possesses both a numerical value with units (such as meters, $\text{m}$) and a sign ($+$ or $-$) indicating its direction relative to the origin.
   • For example, $x = +5\text{ m}$ indicates that the object is located 5 meters to the right (or north) of the origin, whereas $x = -3\text{ m}$ means it is located 3 meters to the left (or south).

   Distance Traveled

   When an object moves from one position to another, we can measure how much ground it has covered along its trajectory.

   Distance ($d$) is a scalar quantity that represents the total length of the actual path traveled by an object, regardless of direction.

   • Scalar nature: Distance has magnitude (size and unit) only; it does not have a direction.
   • Non-negative value: Because path length accumulates with any motion, distance is always greater than or equal to zero ($d \ge 0$).
   • SI Unit: The standard unit of distance in the International System of Units (SI) is the meter ($\text{m}$).

   If an object moves 4 meters east and then turns around and walks 4 meters west, the total distance traveled is $4\text{ m} + 4\text{ m} = 8\text{ m}$.

   Displacement

   Rather than asking how far an object traveled in total, we often need to know how far out of place it is from where it started.

   Displacement ($\Delta x$) is a vector quantity that represents the straight-line change in position of an object from its initial position to its final position.

   In one dimension, displacement is calculated as:

   $$\Delta x = x_f - x_i$$

   • $x_i$ is the initial position.
   • $x_f$ is the final position.
   • $\Delta$ (delta) represents "change in".

   Key Properties of Displacement:

   • Vector nature: Displacement has both magnitude and direction. In one-dimensional motion, direction is indicated by an algebraic sign ($+$ or $-$).
   • Significance of sign: A positive displacement ($\Delta x > 0$) means overall motion in the positive direction; a negative displacement ($\Delta x < 0$) indicates net motion in the negative direction.
   • Zero displacement: If an object returns to its starting point ($x_f = x_i$), its displacement is zero ($\Delta x = 0$), regardless of the total path length traveled.

   Key Differences and Practical Example

   Understanding the distinction between distance and displacement is essential for kinematic analysis:

   • Path Dependence: Distance depends on the entire trajectory taken, while displacement depends only on the endpoints ($x_i$ and $x_f$).
   • Magnitude Relationship: The magnitude of displacement is always less than or equal to the distance traveled: $|\Delta x| \le d$. They are equal only when an object moves strictly in a single direction along a straight line without reversing.

   Worked Example

   A runner starts at position $x_i = +2\text{ m}$, runs rightward to $x = +10\text{ m}$, and then turns around and runs leftward to stop at $x_f = +4\text{ m}$.

   • Distance: The runner covers $10\text{ m} - 2\text{ m} = 8\text{ m}$ forward, plus $10\text{ m} - 4\text{ m} = 6\text{ m}$ backward. Total distance $d = 8\text{ m} + 6\text{ m} = 14\text{ m}$.
   • Displacement: $\Delta x = x_f - x_i = (+4\text{ m}) - (+2\text{ m}) = +2\text{ m}$.

   Although the runner covered a total distance of 14 meters, their net displacement was only 2 meters in the positive direction.