Chapter 2: Linear Motion – One Dimensional Kinematics

Linear Motion in One Dimension

Introduction to Classical Mechanics and Kinematics

Linear motion is a foundational concept in classical mechanics, focusing on the movement of objects along a straight line. Kinematics is the branch of physics that describes motion without considering its causes.

• Classical Mechanics: Studies the motion of macroscopic objects.

• Kinematics: Describes motion (position, velocity, acceleration) without reference to forces.

• One-Dimensional Motion: Motion along a straight line, typically described using a single coordinate axis.

Position, Displacement, and Distance

Coordinates and Position

To describe motion in one dimension, a coordinate system is established, usually with the origin at x = 0. Motion to the right or upward is considered positive, while motion to the left or downward is negative.

• Position (x): The location of an object along the coordinate axis.

Displacement

Displacement is a vector quantity representing the change in position of an object.

• Definition: The difference between final and initial positions.

• Formula: $\Delta x = x_2 - x_1$

• Properties: Has both magnitude and direction; SI unit is meter (m).

• Example: If a race car moves from 19 m to 277 m, $\Delta x = 277\,\text{m} - 19\,\text{m} = 258\,\text{m}$.

Distance

Distance is a scalar quantity representing the total length of the path traveled, regardless of direction.

• Always positive

• Example: Odometer reading in a car

• Comparison: Displacement considers direction; distance does not.

Displacement vs. Distance Example

• A cyclist rides 3 km west and then 2 km east. Displacement: $-3\,\text{km} + 2\,\text{km} = -1\,\text{km}$ (net change in position) Distance: $3\,\text{km} + 2\,\text{km} = 5\,\text{km}$ (total path length)

Velocity and Speed

Average Velocity

Average velocity is the rate of change of displacement over time. It is a vector quantity.

• Formula: $v_{\text{average},x} = \frac{x_2 - x_1}{t_2 - t_1} = \frac{\Delta x}{\Delta t}$

• Units: m/s (meters per second)

• Direction: Follows the direction of displacement

• Example: If a truck moves from 277 m to 19 m in 10 s, $v_{\text{average},x} = \frac{19 - 277}{16.0 - 6.0} = \frac{-258}{10} = -25.8\,\text{m/s}$

Average Speed

Average speed is the total distance traveled divided by the time interval. It is a scalar quantity.

• Formula: $v_{\text{average}} = \frac{\text{distance}}{\text{time interval}}$

• Units: m/s

• Always positive

Constant Velocity

An object has constant velocity if its average velocity is the same for any segment of its motion.

• Motion diagram: Shows equal spacing between positions at equal time intervals.

Graphical Analysis of Motion

Position-Time (x-t) Graphs

Graphs of position versus time are useful for visualizing motion.

• Slope of x-t graph: Indicates velocity ($v = \frac{\Delta x}{\Delta t}$)

• Straight line: Constant velocity

• Steeper slope: Greater velocity magnitude

• Negative slope: Negative velocity (motion in opposite direction)

Interpreting x-t Graphs

Comparing multiple objects on x-t graphs allows analysis of their velocities and displacements.

• Object with steepest slope: greatest velocity

• Object ending at highest x: greatest positive displacement

• Object ending at lowest x: greatest negative displacement

Equation for Constant-Velocity Linear Motion

• General equation: $x = x_0 + v_x t$

• Where: $x_0$ is initial position, $v_x$ is constant velocity, $t$ is time

Instantaneous Velocity and Speed

Instantaneous Velocity

Instantaneous velocity is the velocity of an object at a specific instant in time.

• Definition: Slope of the tangent to the x-t graph at a given time

• Instantaneous speed: Magnitude of instantaneous velocity

Acceleration

Average Acceleration

Acceleration is the rate of change of velocity with respect to time. It is a vector quantity.

• Formula: $a_{\text{average},x} = \frac{v_{2x} - v_{1x}}{t_2 - t_1} = \frac{\Delta v_x}{\Delta t}$

• Units: m/s2

• Occurs when: Velocity changes in magnitude, direction, or both

Velocity and Acceleration Relationship

• If velocity and acceleration are in the same direction: object speeds up

• If velocity and acceleration are in opposite directions: object slows down

• If velocity is zero but acceleration is nonzero: object is changing direction

Graphical Analysis: Velocity-Time (v-t) Graphs

• Slope of v-t graph: Indicates acceleration ($a = \frac{\Delta v}{\Delta t}$)

• Positive slope: Positive acceleration

• Negative slope: Negative acceleration

Interpreting x-t and v-t Graphs

Type of graph	x-t graph	v-t graph
Value of graph	Coordinate x at time t	Velocity v at time t
Slope of graph	Velocity v at time t	Acceleration a at time t
Changes in slope	Acceleration a at time t	Whether acceleration is changing

Summary Table: Key Equations

Quantity	Equation	Units
Displacement	$\Delta x = x_2 - x_1$	m
Average velocity	$v_{\text{average},x} = \frac{\Delta x}{\Delta t}$	m/s
Average speed	$v_{\text{average}} = \frac{\text{distance}}{\text{time interval}}$	m/s
Average acceleration	$a_{\text{average},x} = \frac{\Delta v_x}{\Delta t}$	m/s2
Constant velocity motion	$x = x_0 + v_x t$	m

Practice and Conceptual Questions

• Quick Quiz: Which conditions cause average velocity to be less than average speed? (Answer: When the object reverses direction during the interval.)

• Example: Calculate displacement, distance, average velocity, and average speed for various scenarios.

Additional info: These notes cover the essential concepts of Chapter 2: Linear Motion, including definitions, equations, graphical analysis, and practical examples relevant for introductory college physics.