Physics Foundations: Units, Measurement, and 1D Kinematics

Math Review and Scientific Notation

Working with Powers of Ten and Significant Figures

Physics relies heavily on expressing quantities using scientific notation and understanding significant figures. This allows for clear communication of very large or small numbers and the precision of measurements.

• Scientific Notation: Numbers are written as a product of a coefficient and a power of ten, e.g., $8.69 \times 10^5$.

• Significant Figures: The digits in a measurement that are known with certainty plus one estimated digit. Calculations should reflect the correct number of significant figures.

• Examples:

  • Convert $2.5 \times 10^{-1}$ to decimal: $0.25$

  • Express $286.6$ mm in scientific notation: $2.866 \times 10^{-1}$ m

Metric (SI) Prefixes

SI prefixes are used to simplify the expression of very large or small quantities. Each prefix represents a specific power of ten.

Prefix Name	Prefix Symbol	Scientific Notation	Exponential Notation	Name	Decimal Form
tera	T	1.0E+12	$10^{12}$	Trillion	1,000,000,000,000
giga	G	1.0E+09	$10^{9}$	Billion	1,000,000,000
mega	M	1.0E+06	$10^{6}$	Million	1,000,000
kilo	k	1.0E+03	$10^{3}$	Thousand	1,000
centi	c	1.0E-02	$10^{-2}$	Hundredth	0.01
milli	m	1.0E-03	$10^{-3}$	Thousandth	0.001
micro	μ	1.0E-06	$10^{-6}$	Millionth	0.000001
nano	n	1.0E-09	$10^{-9}$	Billionth	0.000000001

Example: $3 \times 10^6$ volts = 3 megavolts (MV)

Intro to Physics Units

Importance of Units and Dimensional Analysis

Units are essential in physics for clarity and accuracy. Dimensional analysis uses the "magic one"—multiplying by conversion factors equal to one—to convert between units.

• Magic One: Multiplying by a conversion factor that equals one does not change the value, only the units.

• Example: To convert 6.5 dozen to objects: $6.5 \text{ dozen} \times \frac{12 \text{ objects}}{1 \text{ dozen}} = 78 \text{ objects}$

• Example: To convert miles to inches: $1 \text{ mile} = 5280 \text{ ft} \times \frac{12 \text{ in}}{1 \text{ ft}} = 63,360 \text{ in}$

Measurement and Uncertainty

Precision, Accuracy, and Uncertainty

All measurements have some degree of uncertainty. Understanding the difference between accuracy and precision is crucial for interpreting results.

• Accuracy: How close a measurement is to the true value.

• Precision: How repeatable measurements are, regardless of their closeness to the true value.

• Estimated Uncertainty: Expressed as $\pm$ a value, e.g., $5.2 \pm 0.1$ cm.

Example: $10.7$ could mean $10.5$ to $10.9$, while $10.70$ could mean $10.68$ to $10.72$.

Models, Theories, Laws, and Scientific Inquiry

Definitions and Roles in Physics

Physics uses models, theories, and laws to describe and predict natural phenomena.

• Model: A simplified representation to help visualize and understand complex systems (e.g., computer simulations of black holes).

• Theory: A well-tested, broad explanation of phenomena (e.g., quantum mechanics).

• Law: A concise statement describing a fundamental relationship (e.g., conservation of energy).

• Hypothesis: A tentative explanation to be tested.

• Evidence: Data or observations supporting or refuting a hypothesis or theory.

• Scientific Inquiry: The systematic process of investigating scientific questions.

Order of Magnitude and Estimation

Rapid Estimating and Approximations

Order of magnitude estimates help quickly assess the scale of a quantity and check the plausibility of results.

• Order of Magnitude: The power of ten closest to a value.

• Approximations: Useful for quick calculations and sanity checks.

• Example: Estimating the thickness of a textbook page by dividing the total thickness by the number of pages.

Mechanics: Kinematics and Dynamics

Key Definitions

Mechanics is the branch of physics dealing with motion and its causes. Kinematics describes motion, while dynamics explains why motion occurs.

• Mechanics: Study of motion, forces, and energy.

• Kinematics: Describes how objects move (position, velocity, acceleration).

• Dynamics: Explains why objects move (forces and interactions).

• Frame of Reference: The perspective from which motion is measured (e.g., a train or the ground).

• Cartesian Coordinates: The x-y (and z) coordinate system for specifying positions in space.

1D Motion / Kinematics

Describing Motion in One Dimension

One-dimensional kinematics focuses on motion along a straight line, using position, velocity, and acceleration.

• Position (x, y, z): The location of an object in space, often in meters (m).

• Distance: The total length of the path traveled, regardless of direction.

• Displacement: The straight-line change in position, including direction.

• Speed: The magnitude of how fast an object moves ($\text{speed} = \frac{\text{distance}}{\text{time}}$).

• Velocity: The rate of change of position with direction ($\vec{v} = \frac{\Delta \vec{x}}{\Delta t}$).

• Acceleration: The rate of change of velocity ($a = \frac{\Delta v}{\Delta t}$).

Example: A drag race is a classic example of 1D motion, where cars accelerate along a straight track.

• Distance vs. Displacement: Driving to a destination and back covers a large distance, but the displacement could be zero if you return to your starting point.

Summary Table: Key Kinematics Quantities

Quantity	Symbol	Definition	SI Unit
Position	x, y, z	Location in space	meter (m)
Distance	d	Total path length	meter (m)
Displacement	Δx	Straight-line change in position	meter (m)
Speed	v	Rate of distance change	m/s
Velocity	\vec{v}	Rate of displacement change	m/s
Acceleration	a	Rate of velocity change	m/s²

Additional info: The notes also reference the importance of using correct units, the difference between models, theories, and laws, and the process of scientific inquiry, all foundational to the study of physics.