Physics Study Guide: Chapters 1–5 (Kinematics, Vectors, and Motion)

Significant Figures and Scientific Notation

Understanding Scientific Notation

Scientific notation is a method used to express very large or very small numbers in a compact form. It consists of a prefactor (a number between 1 and 10) multiplied by a power of ten. This allows for easier comparison and calculation with numbers that are otherwise difficult to interpret.

• Prefactor: The number between 1 and 10 in scientific notation.

• Exponent: Indicates how many times the prefactor is multiplied or divided by ten.

• Example: $0.0035 \times 10^4$ vs. $3.50 \times 10^{-5}$ — scientific notation helps determine which is larger.

Key Point: Always ensure the prefactor is between 1 and 10 for proper scientific notation.

Unit Conversion and Dimensional Analysis

Converting Units

Unit conversion is essential for solving physics problems. Dimensional analysis involves writing conversion factors as fractions and multiplying them to cancel units appropriately.

• Conversion Factor: A ratio used to convert from one unit to another (e.g., $1 \text{ inch} = 2.54 \text{ cm}$).

• Method: Write facts as fractions and multiply, inverting the conversion factor if necessary to cancel units.

• Example: To convert 10 inches to centimeters: $10 \text{ in} \times \frac{2.54 \text{ cm}}{1 \text{ in}} = 25.4 \text{ cm}$

Displacement, Average Velocity, and Average Acceleration

Definitions and Calculations

These quantities describe motion and are often referenced in graphs of position, velocity, or acceleration versus time. Understanding the difference between average and instantaneous values is crucial.

• Displacement ($\Delta x$): The change in position, $\Delta x = x_f - x_i$.

• Average Velocity ($v_{ave}$): $v_{ave} = \frac{x_f - x_i}{t_f - t_i}$

• Average Acceleration ($a_{ave}$): $a_{ave} = \frac{v_f - v_i}{t_f - t_i}$

• Instantaneous vs. Average: When a value is constant, average and instantaneous values are equal.

Example: For constant velocity, the position vs. time plot is a straight line.

Graph Reading and Vector Directions

Interpreting Position vs. Time Graphs

Reading graphs is fundamental in physics. The slope of a position vs. time graph represents the average velocity. The value, sign, and slope of the graph provide information about the motion.

• Slope: Indicates the rate of change of position (velocity).

• Value/Amount: The position at a given time.

• Algebraic Sign: Indicates direction (positive or negative velocity).

• Constant Slope: Means constant velocity.

• Horizontal Line: Indicates zero velocity.

Equation for Average Velocity:

$v_{ave} = \frac{\Delta x}{\Delta t} = \frac{x_2 - x_1}{t_2 - t_1}$

Example: Calculating the slope between two points on a position vs. time graph gives the average velocity. The sign of the slope indicates direction, not whether velocity is increasing or decreasing.

Vector Algebra

Component and Graphical Methods

Vector algebra involves decomposing vectors into components and adding or subtracting them. Angles are described relative to axes or cardinal directions, and graphical methods such as the tail-to-tip method are used for vector addition.

• Component Method: Add/subtract vectors by their x- and y-components.

• Angle Description: Angles are measured counterclockwise from the positive x-axis unless specified otherwise.

• Cardinal Directions: North as positive y-axis, angles specified as degrees West of North, etc.

• Graphical Method: Tail-to-tip method for adding vectors; translation does not change components.

Example: Adding vectors in lab by calculating components and drawing them without a ruler or protractor.

Constant Acceleration Problems (1D and 2D)

Solving Kinematics Equations

Problems involving constant acceleration require careful use of kinematic equations and attention to vector components and implied variables. Conceptual questions may involve the relationship between velocity and acceleration, especially in free-fall scenarios.

• Kinematic Equations:

  • $v_y = v_{0y} + a_y t$

  • $y = y_0 + v_{0y} t + \frac{1}{2} a_y t^2$

  • $v_y^2 = v_{0y}^2 + 2 a_y \Delta y$

  • $v_x = v_{0x}$

  • $x = x_0 + v_{0x} t$

• Common Mistakes: Not using the correct component, forgetting implied variables (e.g., $v = 0$), and not carefully writing equations.

Example: Solving for unknowns in 2D motion may require using one set of equations to find variables needed in another set.

Additional info: These notes expand on brief points to provide academic context and examples for each topic, ensuring completeness and clarity for exam preparation.