Physics Study Guide: Significant Figures, Units, Vectors, Kinematics

A logical progression through physics begins with classical concepts and moves toward modern topics. Starting with the fundamentals of measurement—significant figures, units, and dimensional analysis—provides a solid foundation for later subjects such as vectors, kinematics, dynamics, and beyond.

Significant Figures Rules

When adding or subtracting quantities, the result must retain the same number of decimal places as the measurement with the fewest decimal places. For multiplication or division, the answer is limited to the same number of significant figures as the measurement with the least significant figures. Square‑root calculations follow the same rule as multiplication: the final answer carries the same number of significant figures as the original value. Scientific notation is useful for displaying the correct number of significant figures, especially when rounding creates trailing zeros.

International System of Units (SI)

The SI system defines seven base units: length, mass, time, electric current, thermodynamic temperature, luminous intensity, and amount of substance. Derived units combine these basics (e.g., density = mass/volume, force = mass × acceleration). The United States also uses a customary system with units such as feet, slugs, and seconds. The SI system was formally established in 1960 and includes twenty prefixes; notably, the kilogram is the only base unit that incorporates a prefix in its name.

Unit Prefixes

Prefixes such as pico (10⁻¹²) and giga (10⁹) indicate the magnitude of a measurement without writing out the full number. They are typically spaced in multiples of a thousand, allowing concise expression of very large or very small values.

Standard Form (Scientific Notation)

Standard form expresses numbers as a decimal between 1 and 10 multiplied by a power of ten. This format simplifies the handling of extreme magnitudes and makes the number of significant figures explicit.

Dimensions and Dimensional Analysis

Physical dimensions—length (L), mass (M), and time (T)—characterize measurable quantities. Dimensional analysis checks whether an equation is plausible by ensuring that the dimensions on both sides match. It can also guide the derivation of relationships between quantities, though it cannot determine dimensionless constants of proportionality.

Unit Conversion

Conversion factors are ratios equal to unity (e.g., 60 s / 1 min = 1) that allow units to be treated algebraically and cancelled during calculations. Multi‑step conversions chain several factors to move between complex unit systems.

Order of Magnitude Calculations

Estimating a quantity to the nearest power of ten provides a quick sense of scale. This “back‑of‑the‑envelope” approach is valuable for rough planning and sanity checks.

Vectors

Vectors possess both magnitude and direction, unlike scalars which have only magnitude. They are commonly drawn as arrows; equal vectors share the same length and orientation regardless of position. Vector addition can be performed graphically (tip‑to‑tail) or by resolving each vector into components using trigonometric relations (adjacent = magnitude × cos θ, opposite = magnitude × sin θ). Vectors must describe the same physical quantity and use the same units to be combined. Negative vectors have identical magnitude but opposite direction, and the magnitude of any vector is always non‑negative.

Kinematics (One‑Dimensional)

Kinematics describes motion without reference to the forces that cause it. In one dimension, motion is confined to a straight line, typically the x‑axis. Key concepts include:

• Position: location relative to a chosen reference point.
• Displacement: change in position, a vector quantity.
• Distance: total path length traveled, a scalar.
• Velocity: rate of change of displacement; a vector.
• Speed: rate of change of distance; a scalar.
• Acceleration: rate of change of velocity; often assumed constant in introductory problems.

Position‑time graphs visualize motion; the slope at any point gives instantaneous velocity, while the overall gradient between two points yields average velocity.

Average Speed vs. Average Velocity

Average speed equals total distance divided by total time. Average velocity equals total displacement divided by total time. For a round trip from point A to B and back to A, the displacement is zero, so the average velocity is zero, while the average speed remains positive.

Instantaneous Velocity and Speed

Instantaneous velocity is the velocity at a specific instant, found as the derivative of position with respect to time (dx/dt). Its magnitude is the instantaneous speed.

Calculating Average and Instantaneous Velocity/Speed

Average velocity between two points can be obtained by drawing a straight line between them on a position‑time graph and measuring its gradient. Instantaneous velocity requires differentiating the position function. For example, if x = −4t + 2t², then vx = dx/dt = −4 + 4t; at t = 2.5 s, vx = 6 m/s.

Acceleration

Acceleration quantifies how quickly velocity changes. Average acceleration is Δv / Δt, while instantaneous acceleration is the derivative of velocity with respect to time (dv/dt) or the second derivative of position (d²x/dt²).