Chapter 2: Motion in One Dimension – Principles & Practice of Physics (Masur)

Motion in One Dimension

Introduction to Kinematics

Kinematics is the branch of physics that describes the motion of objects without considering the causes of motion. In one-dimensional kinematics, we focus on motion along a straight line, using graphical and mathematical methods to analyze position, displacement, speed, velocity, and acceleration.

• Position: The location of an object along a coordinate axis, typically denoted as x.

• Displacement: The change in position of an object, defined as $\Delta x = x_f - x_i$.

• Speed: The rate at which an object covers distance, regardless of direction.

• Velocity: The rate of change of position, including direction; can be average or instantaneous.

• Acceleration: The rate of change of velocity (not covered in detail in these slides).

Example Application: Tracking the position of a train at regular time intervals to analyze its motion.

Scalars and Vectors

Physical quantities in physics are classified as either scalars or vectors. Understanding the distinction is crucial for analyzing motion.

• Scalar: A quantity described by magnitude (number and unit) only. Examples: speed, distance, mass.

• Vector: A quantity described by both magnitude and direction. Examples: displacement, velocity, acceleration.

• Unit Vector: A vector with magnitude one, used to indicate direction (e.g., $\hat{i}$ along the x-axis).

Example: The displacement from an initial to a final position is a vector, while the total distance traveled is a scalar.

Representations of Motion

Motion can be represented in various ways, including:

• Diagrams: Visual sketches showing position at different times.

• Tables: Tabular data listing position or other quantities at specific times.

• Graphs: Plots of position vs. time, velocity vs. time, etc., which reveal patterns and allow for quantitative analysis.

Correlating these representations helps determine kinematic quantities such as displacement, speed, and velocity.

Concepts and Self-Quizzes

Average Speed and Average Velocity

Average speed and average velocity are fundamental concepts in kinematics:

• Average Speed: Total distance traveled divided by the total time interval.

• Average Velocity: Displacement divided by the total time interval; includes direction.

Formulas:

• Average speed: $\text{Average speed} = \frac{\text{distance}}{\Delta t}$

• Average velocity: $v_{x,\text{av}} = \frac{x_f - x_i}{t_f - t_i}$

Example: If a person walks from $x_i = 1.0$ m to $x_f = 4.8$ m in $2.3$ s, the average speed and average velocity are both $1.7$ m/s (since the motion is in one direction).

Graphical Interpretation of Motion

Position vs. time graphs provide insight into the nature of motion:

• Slope of x(t) graph: Represents velocity; a steeper slope means higher speed.

• Horizontal line: Object at rest.

• Upward slope: Motion in positive direction.

• Downward slope: Motion in negative direction.

Self-Quiz Example: An object moving away from the origin at constant speed is represented by a straight, upward-sloping line.

Impossible Motions

Some position vs. time graphs may represent impossible motions, such as an object being in two places at the same time (multiple intersections for a single time value).

Velocity as a Vector

Velocity is a vector quantity and can be expressed using unit vector notation:

• $\vec{v}_{\text{av}} = v_{x,\text{av}} \hat{i} = \frac{\Delta \vec{r}}{\Delta t}$

Where $\Delta \vec{r}$ is the displacement vector and $\hat{i}$ is the unit vector in the x-direction.

Motion at Constant Velocity

Constant Velocity Equations

When an object moves at constant velocity, its position changes linearly with time:

• $v_{x,\text{av}} = v_x$ (velocity is constant)

• $\Delta x = v_x \Delta t$

• $x_f = x_i + v_x \Delta t$

The area under the velocity vs. time graph gives the displacement during the interval.

Worked Example: Constant Velocity

Given a position vs. time graph where $x$ changes from $1.6$ m to $5.2$ m in $6.0$ s:

• Velocity: $v_x = \frac{5.2 - 1.6}{6.0} = 0.60$ m/s

• General position: $x(t) = 1.6 + 0.60 t$

• At $t = 25$ s: $x = 1.6 + 0.60 \times 25 = 16$ m

Summary Table: Scalars vs. Vectors

Quantity	Scalar	Vector
Distance	Yes	No
Displacement	No	Yes
Speed	Yes	No
Velocity	No	Yes

Key Equations and Concepts

• Displacement: $\Delta x = x_f - x_i$

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

• Average speed: $\text{Average speed} = \frac{\text{distance}}{\Delta t}$

• Constant velocity position: $x(t) = x_i + v_x t$

• Velocity as a vector: $\vec{v}_{\text{av}} = \frac{\Delta \vec{r}}{\Delta t}$

Summary

• Motion in one dimension is described using position, displacement, speed, and velocity.

• Scalars have magnitude only; vectors have magnitude and direction.

• Graphs and equations are essential tools for analyzing and predicting motion.

• Average speed and average velocity are distinct; average speed depends on total distance, while average velocity depends on displacement.

• Constant velocity leads to linear position vs. time graphs and horizontal velocity vs. time graphs.