Physics: Uniform Rectilinear Motion (MRU & MRUV)

MRU (Uniform Rectilinear Motion)

• Straight trajectory exemplifies MRU.
• Velocity remains constant in magnitude, direction, and sense in MRU.
• Acceleration is zero in MRU.

MRU Equations

• $x = x_0 + v * (t - t_0)$ shows position as a function of time.
• $v = cte.$ means velocity is constant.
• $a = 0$ signifies no acceleration.

MRU Graphs

• x(t): A straight line represents position over time.
• v(t): A horizontal line represents constant velocity over time.
• a(t): There is no graph for acceleration, as it is zero.

MRUV (Uniformly Varied Rectilinear Motion)

• Straight trajectory is a characteristic of MRUV.
• Constant acceleration in magnitude, direction, and sense defines MRUV.
• Velocity is variable in MRUV.

MRUV Equations

• $x = x_0 + v_0 * (t - t_0) + \frac{1}{2} * a * (t - t_0)^2$ shows position over time with constant acceleration.
• $v = v_0 + a * (t - t_0)$ gives velocity as a function of time and constant acceleration.
• $a = cte.$ indicates constant acceleration.

MRUV Graphs

• x(t): A parabola describes position over time.
• v(t): A straight line represents velocity over time.
• a(t): A horizontal line shows constant acceleration over time.

Vertical Throw

• MRUV characteristics are present
• Vertical straight-line trajectory typifies a vertical throw.
• Acceleration equals -g, approximately -9.8 m/s², acting downward.

Vertical Throw Equations

• $y = y_0 + v_0 * (t - t_0) - \frac{1}{2} * g * (t - t_0)^2$ gives vertical position over time.
• $v = v_0 - g * (t - t_0)$ relates velocity to time with gravitational acceleration.
• $a = -g$ indicates constant downward acceleration due to gravity.

Free Fall

• MRUV characteristics are present
• Vertical straight-line path is followed
• Acceleration equals g, approximately 9.8 m/s², acting downward.
• $v_0 = 0$ means the initial velocity is zero.

Free Fall Equations

• $y = y_0 + v_0 * (t - t_0) + \frac{1}{2} * g * (t - t_0)^2$ defines the vertical position over time.
• $v = v_0 + g * (t - t_0)$ gives velocity as a function of time.
• $a = g$ indicates constant downward acceleration due to gravity.

Algorithmic Complexity: Fundamentals

• Algorithmic complexity measures resources required by an algorithm.
• Time and space are quantified as functions of input size.
• Big O notation expresses algorithmic complexity where n is the input size: $O(f(n))$.
  • This describes an upper bound for how fast the algorithm runs.
  • Big O focuses on the dominant term as $n$ approaches infinity.
• Helps to compare the performance of different algorithms.

Determining Algorithmic Complexity

• First Step: Identify the input size as the parameter determining data processed.
• Second Step: Count elementary operations like arithmetic, comparisons, assignments, and memory access.
• Express those operations as a function of input size considering best, average, and worst-case scenarios.
• Simplify using Big O rules:
  • Drop constants, e.g., $O(2n) \rightarrow O(n)$.
  • Keep the dominant term, e.g., $O(n^2 + n) \rightarrow O(n^2)$.

Common Algorithmic Complexities

• O(1): Constant time (accessing an array element by index).
• O(log n): Logarithmic time (binary search).
• O(n): Linear time (iterating through an array).
• O(n log n): Linearithmic time (merge sort).
• O(n^2): Quadratic time (nested loops).
• O(2^n): Exponential time (trying all possible subsets).
• O(n!): Factorial time (trying all possible permutations).

Why Algorithmic Complexity Matters

• Crucial for performance and scalability.
• Informs decisions about data structures based on performance.
• Allows prediction of runtime/memory usage as input grows.
• Enables identification of bottlenecks for code improvement.

Linear Search Example

• Input size: $n$ (length of the array).
• Operations: Comparison, assignment.
• Worst case: The target is not in the array, requiring $n$ comparisons.
• Complexity: $O(n)$ (linear time).

Algorithmic Complexity: Visual Representation

• O(1) is illustrated by a horizontal line, indicating time doesn't change with input size.
• O(log n) is a slowly rising curve, showing a small increase in time as input grows.
• O(n) is a straight line, indicating a proportional increase in time with input size.
• O(n log n) is a curve rising faster than O(n) but slower than O(n^2).
• O(n^2) shows the impact of quadratically increasing time.
• O(2^n) indicates exponential growth, rising extremely fast with input size.

Matrices

• A matrix A of size m × n is a rectangular array of scalars arranged in m rows and n columns.
• $A = \begin{bmatrix} a_{11} & a_{12} & \dots & a_{1n} \ a_{21} & a_{22} & \dots & a_{2n} \ \vdots & \vdots & \ddots & \vdots \ a_{m1} & a_{m2} & \dots & a_{mn} \end{bmatrix}$ , where $a_{ij}$ represents the element at the i-th row and j-th column.
• If m = n, A is a square matrix of order n.
• If m = 1, A is a row matrix or row vector.
• If n = 1, A is a column matrix or column vector.

Matrix Operations: Addition and Subtraction

• The sum A + B of two matrices A and B of the same size m × n is a matrix m × n with elements $(A + B){ij} = a{ij} + b_{ij}$.
• Subtraction A - B is defined similarly with $(A - B){ij} = a{ij} - b_{ij}$.

Matrix Operations: Scalar Multiplication

• The multiplication of a matrix A by a scalar c, denoted cA, is a matrix obtained by multiplying each element of A by c: $(cA){ij} = c \cdot a{ij}$.

Matrix Operations: Matrix Multiplication

• The product AB of matrices A (m × p) and B (p × n) is a matrix m × n with elements $(AB){ij} = \sum{k=1}^{p} a_{ik}b_{kj}$.
• Matrix multiplication is only defined if the number of columns of the first matrix equals the number of rows of the second matrix.
• Matrix multiplication is not commutative: AB ≠ BA in general.
• So $A(BC) = (AB)C$ (associativity).
• And $A(B+C) = AB + AC$ (left distributivity).
• And $(A+B)C = AC + BC$ (right distributivity).
• Scalar multiplication is $c(AB) = (cA)B = A(cB)$.

Transposition

• The transpose of a matrix A (m × n), denoted Aᵀ, is the matrix n × m with rows and columns swapped: $(A^T){ij} = a{ji}$.

Transposition Properties

• $(A^T)^T = A$
• $(A+B)^T = A^T + B^T$
• $(cA)^T = cA^T$
• $(AB)^T = B^T A^T$

Special Matrices

• Null matrix: all elements are zero.
• Identity matrix I_n: a square matrix of order n with diagonal elements equal to 1 and all other elements equal to 0.
• The properties of a null matrix is that A + 0 = A.
• The properties of the identity matrix are that $AI_n = A$, and $I_mA = A$.
• Diagonal matrix: A square matrix with non-zero diagonal elements.
• Triangular Matrix: A square matrix with all elements below or above the diagonal being zero.
• Symmetric: A square matrix A is symmetric if Aᵀ = A.
• Antisymmetric: A square matrix A is antisymmetric if Aᵀ = -A.

Measurements

• Quantities are expressed with a number and a unit (e.g., 10.6 g).
• SI units are used in science.

SI Units and Symbols

• Mass: Kilogram (kg)
• Length: Meter (m) -Time: Second (s)
• Temperature: Kelvin (K)
• Amount of substance: Mole (mol)
• Electric current: Ampere (A)
• Luminous intensity: Candela (cd)

Prefixes and Multipliers

• Giga (G): $10^9$
• Mega (M): $10^6$
• Kilo (k): $10^3$
• Deci (d): $10^{-1}$
• Centi (c): $10^{-2}$
• Milli (m): $10^{-3}$
• Micro ($\mu$): $10^{-6}$
• Nano (n): $10^{-9}$
• Pico (p): $10^{-12}$
• Femto (f): $10^{-15}$

Volume Measurement

• Volume is not an SI unit but derived from length.
• $1 cm^3 = 1 mL$
• $1 dm^3 = 1 L$

Temperature

• Celsius: Water freezes at $0^\circ C$, boils at $100^\circ C$
• Kelvin: $K = ^\circ C + 273.15$, water freezes at 273.15 K, boils at 373.15 K

Density Calculation

• Definition: Density is calculated by $Density = \frac{mass}{volume}$.
• Formula: $d = \frac{m}{V}$
• Common units: $g/mL$ or $g/cm^3$

Scientific Notation

• Express numbers as $A \times 10^b$.
• Where $A$ is between 1 and 10, and $b$ is an integer.

Scientific Notation Examples

• 5,680 = $5.680 \times 10^3$
• 0.00000772 = $7.72 \times 10^{-6}$

Significant Figures

• Includes all certain digits plus one estimated digit.
• Non-zero digits are always significant.
• Zeros between non-zero digits are significant.
• Zeros before the first non-zero digit are NOT significant.
• Zeros at the end are significant if there's a decimal point.
• Exact numbers have infinite significant figures.

Significant Figures Examples

• 0. 0070 = 2 SF
• 1. 050 = 3 SF
• 2. 230 = 4 SF
• 3. = 3 SF
• 4. 100 = 1 SF

Significant Figures in Calculations

• Multiplication/Division: Result has the fewest significant figures from original data.
• Addition/Subtraction: Result has the fewest decimal places from original data.

Rounding Rules

• If the digit removed is greater than 5, the preceding number increases by 1.
• If the digit removed is less than 5, the preceding number is unchanged.
• If the digit removed is 5, the preceding number is rounded to the nearest even number.

Accuracy vs. Precision

• Accuracy indicates how close a measurement is to the true value.
• Precision indicates how close a set of measurements are to each other.

Visual Representation of Accuracy/Precision

• High accuracy, high precision: darts clustered tightly in the bullseye.
• Low accuracy, high precision: darts clustered tightly, but far from the bullseye.
• Low accuracy, low precision: darts scattered randomly.

Dimensional Analysis for Unit Conversion

• Using conversion factors enables changing units.
• Example: Convert 12.0 inches to meters.
• Use 1 inch = 2.54 cm and 100 cm = 1 m:

$12.0 \text{ in} \times \frac{2.54 \text{ cm}}{1 \text{ in}} \times \frac{1 \text{ m}}{100 \text{ cm}} = 0.305 \text{ m}$

Chemical Reaction

• The transformation of a substance into a new substance.

Chemical Equation

• Employs chemical symbols to represent a chemical reaction.

Features of Chemical Equations

• Equations must represent real reactions.
• Equations must adhere to the Law of Conservation of Mass where all atoms on the left must appear on the right.
• Balancing involves adding coefficients to ensure the atoms on both sides are accounted for.

Guidelines for Writing Equations

• Begin by writing the accurate formulas for reactants and products.
• Present all reactants and products
• Start with the most complex formula and balance by inspection, working towards balancing H and O atoms last.
• Balance polyatomic ions as single units if present on both sides.
• Verify equal elements on both sides of the equation.
• Reduce to smallest possible integers and ensure atom count balance.

Equations Example

• $CH_4 + 2O_2 \rightarrow 2H_2O + CO_2$
• $4NH_3 + 7O_2 \rightarrow 6H_2O + 4NO_2$