Physics Fundamentals: Units, Significant Figures, and Kinematics

Units, Scientific Notation, and Significant Figures

Scientific Notation

Scientific notation is a way to express very large or very small numbers using powers of ten. This is especially useful in physics for clarity and precision.

• Definition: A number is written as a × 10n, where 1 ≤ a < 10 and n is an integer.

• Example: 0.0012 can be written as $1.2 \times 10^{-3}$.

Significant Figures

Significant figures (sig figs) indicate the precision of a measured or calculated quantity. The number of significant figures is determined by the certainty of the measurement.

• Rules:

  • All nonzero digits are significant.

  • Zeros between nonzero digits are significant.

  • Leading zeros are not significant.

  • Trailing zeros after a decimal point are significant.

• Example: The number 0.003010 has 4 significant figures (3, 0, 1, 0).

Rounding and Calculations with Significant Figures

When performing calculations, the result should be rounded to the correct number of significant figures based on the operation:

• Multiplication/Division: The result should have as many significant figures as the value with the fewest significant figures.

• Addition/Subtraction: The result should have the same number of decimal places as the value with the fewest decimal places.

• Example: Multiplying 1.125 m × 0.606 m = 0.68175 m2. Rounded to 2 significant figures: 0.68 m2.

Unit Conversions

Unit conversions are essential in physics to ensure consistency and correctness in calculations.

• Example: Converting kilograms to milligrams:

  • 1 kg = $1 \times 10^6$ mg

  • 171 kg = $1.71 \times 10^8$ mg

• Speed Conversion: To convert km/h to ft/min:

  • 1 km = 1000 m; 1 m = 3.28 ft; 1 h = 60 min

  • Example: 4.50 km/h = 246 ft/min

Basic Kinematics and Motion

Position, Velocity, and Acceleration

Kinematics is the study of motion without considering its causes. The primary quantities are position, velocity, and acceleration.

• Position: The location of an object at a particular time.

• Velocity: The rate of change of position with respect to time.

• Acceleration: The rate of change of velocity with respect to time.

• Example: On a position-versus-time graph, the slope at a point gives the object's velocity at that point.

Free Fall and Acceleration Due to Gravity

Objects in free fall experience constant acceleration due to gravity, directed downward near Earth's surface.

• Standard value: $g = 9.80\ \text{m/s}^2$ (downward)

• Equation for vertical acceleration:

  • When upward is positive: $a_y = -g$, where $g = +9.80\ \text{m/s}^2$

  • When downward is positive: $a_y = g$, where $g = -9.80\ \text{m/s}^2$

• Direction: Acceleration is always directed toward the center of the Earth (downward).

Vector Addition and Magnitude

Vectors have both magnitude and direction. The magnitude of a vector can be found using the Pythagorean theorem if its components are known.

• Example: For vector $\vec{C} = \vec{A} + \vec{B}$, where $\vec{A} = 100\hat{i} + 200\hat{j}$ and $\vec{B} = 200\hat{i} + 800\hat{j}$:

  • $\vec{C} = (100+200)\hat{i} + (200+800)\hat{j} = 300\hat{i} + 1000\hat{j}$

  • Magnitude: $|\vec{C}| = \sqrt{(300)^2 + (1000)^2}$

Sample Table: SI Prefixes and Conversions

Prefix	Symbol	Multiplier
kilo	k	$10^3$
milli	m	$10^{-3}$
micro	μ	$10^{-6}$
nano	n	$10^{-9}$

Additional info:

• Some questions involved converting between units (e.g., kg to mg, km/h to ft/min), which is a fundamental skill in introductory physics.

• Questions on significant figures and scientific notation are essential for reporting measurements and results accurately in physics.

• Basic kinematics concepts such as interpreting position-time graphs and understanding free fall are foundational for further study in mechanics.