Motion in One Dimension: Study Notes for College Physics

Chapter 2: Motion in One Dimension

Describing Position

In physics, position is a fundamental concept used to describe the location of an object in space. For one-dimensional motion, we use coordinate axes to specify position:

• x-axis: Used for horizontal motion. The positive direction is typically to the right.

• y-axis: Used for vertical motion. The positive direction is upward.

Position values:

• x > 0: Position to the right of the origin.

• x < 0: Position to the left of the origin.

• y > 0: Position above the origin.

• y < 0: Position below the origin.

Representing Position Graphically

Motion diagrams and position-versus-time graphs are used to visualize how an object's position changes over time. Each dot on a motion diagram represents the object's position at a specific time.

• Slope of position-time graph: Indicates velocity.

• Displacement ($\Delta x$): Change in position over a time interval.

Formulas:

• $v_{x,1} = \frac{\Delta x_1}{\Delta t_1}$

• $v_{x,2} = \frac{\Delta x_2}{\Delta t_2}$

Uniform Motion

Uniform motion occurs when an object moves with constant velocity, meaning equal displacements occur during equal time intervals.

• Key properties: Speed and direction do not change.

• Position-versus-time graph: Straight line; slope equals velocity.

• Equation: $\Delta x = v_x \Delta t$

Example:

If a car moves at a constant speed of 20 m/s for 5 seconds, its displacement is:

$\Delta x = (20\,\text{m/s})(5\,\text{s}) = 100\,\text{m}$

Non-Uniform Motion

Non-uniform motion occurs when an object's velocity changes, either in magnitude (speed), direction, or both. The position-versus-time graph is curved in this case.

• Instantaneous velocity: The velocity at a specific instant, found by computing the slope of the tangent to the curve at that point.

Example:

If the position graph curves upward, the object is speeding up; if it curves downward, the object is slowing down.

Instantaneous Velocity

Instantaneous velocity is the velocity of an object at a specific instant in time. It is defined as the slope of the position-versus-time graph at that point.

• Formula: $v_x = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t}$

Example:

On a curved position-time graph, the instantaneous velocity increases as the slope becomes steeper.

Concept Application: Graph Analysis

Given a position-time graph, you can determine:

• Where displacement is zero: Points where the position returns to the origin.

• Where speed is zero: Points where the slope of the graph is zero (horizontal tangent).

• Where speed is increasing: Points where the graph's slope is increasing.

• Where speed is decreasing: Points where the graph's slope is decreasing.

Acceleration

Acceleration describes how an object's velocity changes over time. It is defined as the rate of change of velocity.

• Formula: $a_x = \frac{\Delta v_x}{\Delta t}$

• Units: $\text{m/s}^2$ (meters per second squared)

Example:

If a car's velocity increases from 10 m/s to 20 m/s in 5 seconds:

$a_x = \frac{20\,\text{m/s} - 10\,\text{m/s}}{5\,\text{s}} = 2\,\text{m/s}^2$

Summary Table: Key Concepts in One-Dimensional Motion

Concept	Definition	Equation
Displacement	Change in position	$\Delta x = x_f - x_i$
Velocity	Rate of change of position	$v_x = \frac{\Delta x}{\Delta t}$
Instantaneous Velocity	Velocity at a specific instant	$v_x = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t}$
Acceleration	Rate of change of velocity	$a_x = \frac{\Delta v_x}{\Delta t}$

Additional info:

• These notes cover the foundational concepts of kinematics in one dimension, including graphical analysis and the mathematical relationships between position, velocity, and acceleration.

• Further study will include kinematic equations for constant acceleration and problem-solving strategies.