Representing Position and Displacement Notes

Fundamentals of Representing Position

• To accurately describe and represent the position of an object, a coordinate axis must be established.

• The establishment of a coordinate axis requires three specific components:

  • Origin: This is the reference point designated as $0$. On a standard axis, this is labeled as $0\,m$.

  • Direction: The axis must clearly define positive ($+$) and negative ($-$) directions. In a standard horizontal orientation, the positive direction typically points to the right ($+x$) and the negative direction points to the left.

  • Unit: A physical quantity of measurement must be assigned to the axis to provide scale. In these examples, the unit used is the meter ($m$).

• Example of Static Position:

  • Consider an object located on a coordinate axis where the origin is denoted as $0\,m$, the units are meters ($m$), and the positive direction is to the right.

  • If an object is placed two units to the right of the origin, its position is represented as $x = 2\,m$.

Movement: Distance vs. Displacement

• Definition of Displacement: Displacement is defined as the change in position of an object. It is represented by the mathematical formula:

  • $\Delta x = x_f - x_i$

  • In this formula, $\Delta x$ represents the displacement, $x_f$ represents the final position, and $x_i$ represents the initial position.

• Path Independence: A critical characteristic of displacement is that it is independent of the path taken. Displacement only considers the starting point and the ending point, regardless of the route traveled between them.

Mathematical Examples and Scenarios of Displacement

• Scenario 1: Positive Displacement

  • An object starts at an initial position ($x_i$) of $-2\,m$.

  • The object moves to a final position ($x_f$) of $3\,m$.

  • Applying the formula $\Delta x = x_f - x_i$:

    • $\Delta x = 3\,m - (-2\,m)$

    • $\Delta x = 5\,m$

  • The resulting displacement is $5\,m$ in the positive direction.

• Scenario 2: Negative Displacement

  • An object starts at an initial position ($x_i$) of $2\,m$.

  • The object moves to the left to a final position ($x_f$) of $-1\,m$.

  • Applying the formula $\Delta x = x_f - x_i$:

    • $\Delta x = -1\,m - 2\,m$

    • $\Delta x = -3\,m$

  • The resulting displacement is $-3\,m$, indicating movement in the negative direction.

• Scenario 3: Illustrating Path Independence

  • Consider two different paths for an object moving from an initial position ($x_i$) of $0\,m$ to a final position ($x_f$) of $3\,m$.

  • Path A: The object moves in a direct, straight line from $0\,m$ to $3\,m$.

  • Path B: The object moves from $0\,m$, travels forward to a point such as $4\,m$, then moves backward, and finally settles at $3\,m$.

  • Despite the extra distance traveled in Path B, both paths result in the same displacement:

    • $\Delta x = 3\,m - 0\,m = 3\,m$

  • This confirms that $\Delta x$ remains the same ($3\,m$) regardless of the complexity of the path taken between the two points.