Physics Study Guide: Measurement, Units, and Representing Motion

Measurement and Physical Quantities

Physical Quantities and Units

Physics involves the study of natural phenomena using measurements and equations. Every physical quantity (such as mass, length, time) must be described with both a number and a unit.

• Physical Quantity: A property that can be measured (e.g., mass, length).

• Unit: A standard amount used to express a physical quantity (e.g., kilogram, meter).

• Example: Measuring the mass of a box as 5 kg.

For equations to work, all units must be compatible with each other. Groups of compatible units form a system of units. In Physics, the SI system (Système International) is always used.

Quantity	SI	Imperial
Mass	Kilogram [kg]	Pound [lb]
Length	Meter [m]	Foot [ft]
Time	Second [s]	Second [s]
Force	Newton [N]	Foot-pound

Force Equation:

$F = m \times a$

Units: $[\text{kg}] \times [\text{m/s}^2] = [\text{N}]$ (Compatible)

Metric Prefixes and Scientific Notation

Metric Prefixes

Metric prefixes are letters or symbols that go before a base unit to indicate a power of ten. They help express very large or very small quantities efficiently.

• Example: 5 km = 5,000 m

• Common prefixes: kilo- (k, $10^3$), centi- (c, $10^{-2}$), milli- (m, $10^{-3}$), micro- ($\mu$, $10^{-6}$)

Prefix	Symbol	Power of Ten
tera	T	$10^{12}$
giga	G	$10^9$
mega	M	$10^6$
kilo	k	$10^3$
hecto	h	$10^2$
deca	da	$10^1$
base unit	-	$10^0$
deci	d	$10^{-1}$
centi	c	$10^{-2}$
milli	m	$10^{-3}$
micro	μ	$10^{-6}$
nano	n	$10^{-9}$
pico	p	$10^{-12}$

Converting with Prefixes:

• Shifting from a bigger to smaller unit: number becomes larger

• Shifting from a smaller to bigger unit: number becomes smaller

Scientific Notation

Scientific notation is used to write very large or very small numbers in a compact form.

General Format: $A.BC \times 10^n$

• Move decimal to get 1 <= A < 10

• Exponent $n$ is the number of places moved

• If original number > 1, $n$ is positive; if < 1, $n$ is negative

Example: $5,972,000,000,000,000,000,000,000$ kg = $5.972 \times 10^{24}$ kg

Unit Conversion

Converting Units

Physics problems often require converting non-SI units to SI units before using equations. This is done using conversion factors.

Quantity	Conversion Factors / Ratios
Mass	1 kg = 2.2 lb; 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

• Write given and target units

• Write conversion factors/ratios

• Multiply and cancel units as needed

Example: Convert 22 lbs to kg:

$22 \text{ lb} \times \frac{1 \text{ kg}}{2.2 \text{ lb}} = 10 \text{ kg}$

Precision and Significant Figures

Precision in Measurements

Precision is indicated by the number of digits in a measurement. More digits mean higher precision.

• 10 kg: less precision

• 10.27 kg: more precision

Significant figures are the digits in a measurement that matter for precision.

• Leading zeros are not significant

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

• Middle zeros are always significant

Example: 0.013200972000 has 9 significant figures

Rules for Calculations with Significant Figures

• Addition/Subtraction: Round answer to the same decimal places as the least precise value

• Multiplication/Division: Round answer to the same number of significant figures as the least precise value

Vectors and Scalars

Measurement Types

Measurements can be classified as vectors (having magnitude and direction) or scalars (having only magnitude).

Measurement	Quantity	Magnitude?	Direction?	Vector/Scalar
Apple weighs 5kg	Mass	Yes	No	Scalar
Days are 24hr long	Time	Yes	No	Scalar
"It's 60°F outside"	Temperature	Yes	No	Scalar
I pushed with 100N left	Force	Yes	Yes	Vector
I walked for 10m	Length	Yes	No	Scalar
I walked 10m east	Length	Yes	Yes	Vector
I drove at 80mph	Speed	Yes	No	Scalar
I drove 80mph west	Velocity	Yes	Yes	Vector

Representing Motion: Distance and Displacement

Distance vs. Displacement

There are two ways to measure how far something moves:

• Distance (d): Total length traveled, regardless of direction. Scalar quantity.

• Displacement ($\Delta x$): Change in position from initial to final point. Vector quantity (includes direction).

Formulas:

Distance: $d = \text{sum of all lengths traveled}$

Displacement: $\Delta x = x_{\text{final}} - x_{\text{initial}}$

Displacement can be negative, but distance is always positive. In physics, $+$ and $-$ signs indicate direction.

Example Problems

• Find displacement and total distance for various paths.

• Calculate magnitude and direction of total displacement.

Additional info: These notes cover foundational concepts for Ch 01: Representing Motion and are essential for understanding measurement, units, and basic motion in physics.