Ch 01: Representing Motion – Foundations of Measurement in Physics

Ch 01: Representing Motion

Introduction to Units and the S.I. System

Physics is the study of natural phenomena, which relies heavily on measurements and equations. To ensure consistency and accuracy, all physical quantities must be expressed with both a number and a unit. The S.I. (Système International) system is the standard in physics for expressing these quantities.

• Physical Quantities: Examples include mass, length, and time. Each must have a value and a unit (e.g., 5 kg, 10 m).

• Unit Consistency: All units in a physics equation must be compatible for the equation to be valid.

• System of Units: A group of compatible units that work together in equations.

• S.I. Base Units:

  • Mass: kilogram (kg)

  • Length: meter (m)

  • Time: second (s)

  • Force: newton (N)

• Imperial Units: Common in the U.S., e.g., pound (lb) for mass, foot (ft) for length.

Quantity	S.I. Unit	Imperial Unit
Mass	kilogram (kg)	pound (lb)
Length	meter (m)	foot (ft)
Time	second (s)	second (s)
Force	newton (N)	foot-pound

Example Equation: Force = Mass × Acceleration

$F = m \times a$

Units: $[N] = [kg] \times [m/s^2]$

Metric Prefixes

Metric prefixes are used to express very large or very small quantities by attaching a prefix to the base unit. Each prefix represents a specific power of ten.

Prefix	Symbol	Power of 10
tera-	T	$10^{12}$
giga-	G	$10^{9}$
mega-	M	$10^{6}$
kilo-	k	$10^{3}$
hecto-	h	$10^{2}$
deca-	da	$10^{1}$
deci-	d	$10^{-1}$
centi-	c	$10^{-2}$
milli-	m	$10^{-3}$
micro-	μ	$10^{-6}$
nano-	n	$10^{-9}$
pico-	p	$10^{-12}$

• When converting from a larger to a smaller unit, the number becomes larger.

• When converting from a smaller to a larger unit, the number becomes smaller.

Example: $5\ km = 5000\ m$

Steps for Conversion:

1. Identify starting and target prefixes.

2. Count the number of powers of ten between them.

3. Shift the decimal accordingly.

Scientific Notation

Scientific notation is used to express very large or very small numbers in a compact form. The general format is $A.BC \times 10^D$, where $1 \leq A.BC < 10$ and $D$ is an integer.

• To convert to scientific notation, move the decimal to get a number between 1 and 10, and count the places moved for the exponent.

• To convert from scientific notation to standard form, move the decimal according to the exponent.

Example: $299,800,000\ m/s = 2.998 \times 10^8\ m/s$

Unit Conversions

Many physics problems require converting between different units. Always convert to S.I. units before using equations.

• Use conversion factors as fractions to cancel out unwanted units.

• For units with exponents, apply the conversion factor for each exponent.

Quantity	Conversion Factors
Mass	1 kg = 2.2 lbs; 1 lb = 450 g; 1 oz = 28.4 g
Length	1 km = 0.621 mi; 1 ft = 0.305 m; 1 in = 2.54 cm
Volume	1 gal = 3.79 L; 1 mL = 1 cm³; 1 L = 1.06 qt

Steps for Unit Conversion:

1. Write the given and target units.

2. Write conversion factors as fractions to cancel units.

3. Multiply all numbers on top and bottom, then solve.

Counting Significant Figures (Sig Figs)

Significant figures reflect the precision of a measurement. Not all digits in a number are significant.

• Leading zeros are not significant.

• Trailing zeros are significant only if there is a decimal point.

• All non-zero digits and zeros between non-zero digits are significant.

Example: 0.0043 has 2 significant figures; 100.00 has 5 significant figures.

Math with Significant Figures

• Addition/Subtraction: Round the answer to the same number of decimal places as the measurement with the least decimal places.

• Multiplication/Division: Round the answer to the same number of significant figures as the measurement with the least significant figures.

• Mixed Operations: Carry extra digits through intermediate steps, round only the final answer.

Introduction to Vectors and Scalars

Physical quantities are classified as either scalars or vectors:

• Scalar: Has magnitude only (e.g., mass, temperature, time).

• Vector: Has both magnitude and direction (e.g., displacement, velocity, force).

Measurement	Quantity	Vector/Scalar
Apple weighs 5kg	Mass	Scalar
Days are 24hr long	Time	Scalar
It's 60°F outside	Temperature	Scalar
I pushed with 100N left	Force	Vector
I walked for 10 ft	Distance	Scalar
I walked 10 ft east	Displacement	Vector

Displacement vs. Distance

Both displacement and distance measure how far something moves, but they are defined differently:

• Distance (d): The total length of the path traveled (scalar, always positive).

• Displacement (Δx): The straight-line change in position from initial to final point (vector, can be positive or negative).

Example: If you walk 10 m east and then 6 m west, your total distance is 16 m, but your displacement is 4 m east.

Introduction to Velocity and Speed

Velocity and speed both describe how fast something moves, but with important differences:

• Speed: Scalar quantity; always positive; $\text{Speed} = \frac{\text{distance}}{\text{time}}$

• Velocity: Vector quantity; can be positive or negative; $\text{Velocity} = \frac{\text{displacement}}{\text{time}}$

Example: If you jog 15 m in 2 s, then 9 m backwards in another 2 s, your total distance is 24 m, but your displacement is 6 m.

Solving Constant and Average Velocity Problems

Average velocity is measured between two points and is calculated as:

$v_{avg} = \frac{\Delta x}{\Delta t}$

• For constant velocity (no acceleration), this is the only equation needed.

• Problems may involve solving for any variable given the others.

Constant Velocity with Multiple Parts

When an object moves with different constant velocities in different intervals, solve each part separately and combine results:

1. Draw a diagram and list variables for each interval.

2. Write equations for each interval.

3. Solve for total distance, displacement, or average velocity as needed.

Introduction to Acceleration

Acceleration describes how quickly velocity changes. It is always a vector and has units of $m/s^2$.

$a = \frac{\Delta v}{\Delta t}$

• Acceleration can result from changes in the magnitude or direction of velocity.

• There is no scalar equivalent for acceleration.

Example: If your car increases its velocity from 10 m/s to 30 m/s in 4 s, the acceleration is:

$a = \frac{30\ m/s - 10\ m/s}{4\ s} = 5\ m/s^2$

Additional info: This guide covers foundational measurement concepts essential for all subsequent topics in physics, including the use of S.I. units, metric prefixes, scientific notation, unit conversions, significant figures, and the distinction between scalars and vectors. These skills are critical for solving physics problems accurately and consistently.