Kinematics in One Dimension: Study Notes

Kinematics in One Dimension

Introduction

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

Uniform Motion

Definition and Characteristics

Uniform motion refers to motion along a straight line at a constant, unvarying speed. The simplest possible motion is uniform motion, where the object's position changes by equal amounts in equal time intervals.

• Uniform motion occurs if and only if the position-versus-time graph is a straight line.

• The displacement between successive frames is constant.

• The slope of the position-versus-time graph represents the velocity.

Mathematical Representation

• Average velocity for one-dimensional motion is given by: $v_{avg} = \frac{\Delta x}{\Delta t}$ (for horizontal motion) $v_{avg} = \frac{\Delta y}{\Delta t}$ (for vertical motion)

• The SI unit of velocity is meters per second (m/s).

• On a position-versus-time graph, the average velocity is the slope of the line.

Uniform-Motion Model

• Model the object as a particle moving in a straight line at constant speed.

• Mathematically: $v_s = \frac{\Delta s}{\Delta t}$ $s_f = s_i + v_s \Delta t$

• Limitations: The model fails if the particle has a significant change of speed or direction.

Relating Position and Velocity Graphs

Example: Velocity from Position Graph

Given a position-versus-time graph, the velocity during each interval can be determined by measuring the slope of the line.

• For a displacement $\Delta x = -4.0\,\text{m}$ over $\Delta t = 2.0\,\text{s}$: $v_x = \frac{-4.0\,\text{m}}{2.0\,\text{s}} = -2.0\,\text{m/s}$

• If the position does not change ($\Delta x = 0$), then $v_x = 0$.

• For a displacement $\Delta x = 10.0\,\text{m}$ over $\Delta t = 2.0\,\text{s}$: $v_x = \frac{10.0\,\text{m}}{2.0\,\text{s}} = 5.0\,\text{m/s}$

On the velocity-versus-time graph, these velocities are represented as horizontal segments corresponding to each time interval.

Scalars and Vectors

Definitions

• Distance: A scalar quantity representing the total length of the path traveled, independent of direction.

• Displacement: A vector quantity equal to the final position minus the initial position; it has both magnitude and direction.

• Speed ($v$): A scalar quantity indicating how fast an object is moving; always positive and independent of direction.

• Velocity: A vector quantity that includes both magnitude and direction. In one dimension, direction is specified by the sign (+ or –).

Comparison Table: Scalars vs. Vectors

Quantity	Type	Direction?
Distance	Scalar	No
Displacement	Vector	Yes
Speed	Scalar	No
Velocity	Vector	Yes

Example

If a car travels 4 meters to the left, then 10 meters to the right, its total distance is 14 meters, but its displacement is 6 meters to the right.

Key Equations

• Average velocity: $v_{avg} = \frac{\Delta x}{\Delta t}$

• Position as a function of time (uniform motion): $x_f = x_i + v \Delta t$

Summary

• Uniform motion is characterized by constant velocity and a straight-line position-versus-time graph.

• Velocity is the slope of the position-versus-time graph.

• Distance and speed are scalars; displacement and velocity are vectors.

• Direction in one dimension is indicated by the sign of displacement or velocity.

Additional info: These notes cover the foundational concepts of kinematics in one dimension, suitable for introductory college physics courses. Later topics such as acceleration, graphical analysis, and kinematic equations are typically covered in subsequent sections.