Motion in One Dimension: Position, Displacement, and Vectors

Motion in One Dimension

From Reality to Model

Kinematics is the branch of physics that describes motion mathematically, without considering its causes. To analyze motion, we track an object's position at different times. If the position changes, the object is in motion; if not, it is at rest. The process of converting real-world observations into quantitative models is fundamental in physics, using tables, graphs, and mathematical functions.

• Position: The location of an object relative to a chosen origin along a reference axis.

• Displacement: The change in position, represented by an arrow from the initial to the final position.

• Modeling: Data from real-world events are abstracted into graphs and tables for analysis.

Position and Displacement

Position is measured along a reference axis (usually the x-axis), and can be positive or negative depending on the direction. Displacement is the difference between the final and initial positions, and is independent of the choice of origin.

• Position (x): The coordinate along the reference axis.

• Displacement (Δx): $\Delta x = x_f - x_i$

• Distance: The total length traveled, always positive.

• Example: If you walk from point P to Q and back, your displacement is zero, but your distance traveled is twice the length between P and Q.

Representing Motion

Motion can be represented graphically as position-versus-time graphs. These graphs show how position changes over time and can be interpolated to form continuous curves.

• Position-versus-time graph: Shows position (vertical axis) as a function of time (horizontal axis).

• Slope: Indicates speed; steeper slopes mean higher speed.

• Example: A straight line indicates constant velocity; a curve indicates changing velocity.

Average Speed and Average Velocity

Average speed is the total distance traveled divided by the time interval. Average velocity is the displacement divided by the time interval, and can be positive or negative depending on direction.

• Average Speed: $\text{Average speed} = \frac{\text{distance traveled}}{\text{time interval}}$

• Average Velocity: $v_{x,\text{av}} = \frac{\Delta x}{\Delta t}$

• Example: If you walk forward and then backward, your average speed reflects total distance, while average velocity reflects net displacement.

Scalars and Vectors

Physical quantities are classified as scalars or vectors. Scalars are described by magnitude and unit only, while vectors require both magnitude and direction.

• Scalar: Quantity with magnitude and unit (e.g., temperature, distance).

• Vector: Quantity with magnitude, unit, and direction (e.g., displacement, velocity).

• Unit Vector: Defines direction; for the x-axis, $\hat{\imath}$.

• Vector Notation: $\mathbf{b} = b_x \hat{\imath}$

Position and Displacement Vectors

Position and displacement can be represented as vectors. The position vector points from the origin to the object's location, and the displacement vector points from the initial to the final position.

• Position Vector: $\mathbf{r} = x \hat{\imath}$

• Displacement Vector: $\Delta \mathbf{r} = \mathbf{r}_f - \mathbf{r}_i = (x_f - x_i) \hat{\imath}$

• Magnitude: $|\Delta \mathbf{r}| = |\Delta x|$

• Vector Addition: To add vectors, place the tail of the second at the tip of the first.

• Vector Subtraction: To subtract, reverse the direction of the vector being subtracted and add.

Velocity as a Vector

Velocity is a vector, defined as the rate of change of position with respect to time. For motion in one dimension, the velocity vector points in the direction of displacement.

• Average Velocity Vector: $\mathbf{v}_{\text{av}} = \frac{\Delta \mathbf{r}}{\Delta t}$

• Instantaneous Velocity: $\mathbf{v} = \frac{d\mathbf{r}}{dt}$

• Speed: The magnitude of velocity, always positive.

Motion at Constant Velocity

If an object moves at constant velocity, its position-versus-time graph is a straight line, and its velocity-versus-time graph is a horizontal line.

• Constant Velocity Equation: $\Delta x = v_x \Delta t$

• Position at Time t: $x_f = x_i + v_x \Delta t$

• Area Under Velocity Curve: Represents displacement.

Instantaneous Velocity

Instantaneous velocity is the velocity at a specific instant, found by taking the derivative of position with respect to time.

• Instantaneous Velocity: $v_x = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt}$

• Graphical Interpretation: The slope of the tangent to the position-versus-time curve at a given instant.

Summary Table: Symbols and Their Meaning

Symbol	Meaning
t	Clock reading (instant in time)
Δt	Time interval ($t_f - t_i$)
x	x coordinate (position)
d	Distance between two points ($|x_1 - x_2|$)
$\hat{\imath}$	Unit vector along x axis
$\mathbf{b}$	Vector ($b_x \hat{\imath}$)
$\Delta \mathbf{r}$	Displacement vector
$\mathbf{r}$	Position vector
$\mathbf{v}_{\text{av}}$	Average velocity vector
$\mathbf{v}$	Instantaneous velocity vector

Relevant Images

The following images visually reinforce the concept of displacement and position vectors, as discussed above: