Distance and Dsiplacement

1. Fundamental Definitions

• Distance ($d$): A scalar quantity that represents the total length of the path traveled by an object. It accounts for every step taken, regardless of the direction. Because it is a scalar, it only has magnitude and no direction.

• Displacement ($\Delta x$): A vector quantity that represents the object's overall change in position. It is the straight-line distance from the starting point to the ending point, including the direction. Because it is a vector, it has both magnitude and direction.

2. How to Draw Them

• Drawing Distance: To visualize distance, draw a line follows the exact trajectory of the object. If the object moves in a zigzag or circular path, the line should reflect that entire path.

• Drawing Displacement: To visualize displacement, draw a single straight arrow (a vector) pointing from the starting position ($x<em>i$) directly to the final position ($x</em>f$).

  • The tail of the arrow is placed at the starting point.

  • The head (tip) of the arrow is placed at the ending point.

  • Example: If a person walks $5 \text{ m}$ East and then $5 \text{ m}$ North, the distance is the two-segment path, while the displacement is the diagonal arrow connecting the start to the end.

3. Computation and Mathematics

• Calculating Distance: Simply sum the lengths of all individual segments of the journey.

  • $d<em>{total} = d</em>1 + d<em>2 + d</em>3 + …$

• Calculating Displacement in 1D: If the motion is along a single axis (e.g., just left and right), subtract the initial position from the final position.

  • $\Delta x = x<em>f - x</em>i$

• Calculating Displacement in 2D: For motion involving changes in two dimensions (like North and East), use the Pythagorean theorem if the segments are perpendicular.

  • Magnitude: $| \Delta x | = \sqrt{a^2 + b^2}$

  • Where $a$ and $b$ are the horizontal and vertical components of the movement.

• Comparison Example: If you run one full lap around a $400 \text{ m}$ track:

  • Distance = $400 \text{ m}$

  • Displacement = $0 \text{ m}$ (since you returned to your starting position).