Regular Expressions (Regex)

Regular Expressions

• Regular expressions (Regexp, or Regex) are character strings that serve to describe sets of strings according to specific syntactic rules.
• They are primarily used in software development.

Applications

• Testing if a string matches a search pattern: Search patterns are templates that must meet certain criteria.
• Replacing strings that match a search pattern: All occurrences of a pattern in a string could be replaced with another string.
• Extracting substrings that match a search pattern: This finds all parts of a string that match a specific pattern and returns them.

Syntax

Characters

Quantifiers

Groups

Examples

• **
• The exact syntax and function support may vary depending on the programming language or tool used.

Economics

• Economics is the study of how societies use scarce resources to produce valuable goods and services and how they distribute them among different individuals.

Key Ideas in Economics

• Goods are scarce
• Society must use its resources efficiently

Branches of Economics

• Economics is divided into two main branches:
• Microeconomics deals with the behavior of individual entities such as markets, firms, and households.
• Macroeconomics deals with the overall performance of the economy.

The Logic of Economics

• Economists use the scientific approach to understand economic events. This involves:
• Observation of economic phenomena
• Formulating theories
• Testing theories

Common Errors in Economic Reasoning

• The post hoc fallacy: Assuming that because one event occurred before another, the first caused the second.
• Failure to hold everything else constant: When analyzing the impact of a variable, it is important to hold all other variables constant.
• The fallacy of composition: Assuming that what is true for one part of the system is also true for the whole.

Positive vs. Normative Analysis

• Positive economics deals with facts and behavior.
• Normative economics involves value judgments.

Central Questions of Economics

• What goods and services are produced and in what quantities?
• How are these goods produced?
• For whom are these goods produced?

Types of Economies

• Market economy: Decisions are made primarily in markets.
• Planned economy: The government makes most of the economic decisions.
• Mixed economy: A combination of market and planned economies.

Technological Possibilities of Society

• Inputs are commodities or services that are used to produce goods and services.
• Products are the various goods and services that result from the production process.

The Production Possibility Frontier (PPF)

• The PPF shows the maximum amounts of production that an economy can obtain, given its technological knowledge and the amount of inputs available.

Applications of the PPF

• Efficiency: To produce the maximum quantity of goods and services with the available resources
• Opportunity cost: The value of the good or service that must be given up to obtain another.
• Economic growth: An outward shift of the PPF.

Momentum and Collisions

Linear Momentum

• Linear momentum is the product of the mass and velocity

Definition of Linear Momentum

• The linear momentum $\vec{p}$ of a particle or an object that can be modeled as a particle of mass $m$ moving with a velocity $\vec{v}$ is defined to be the product of the mass and velocity:

  $\qquad \vec{p} \equiv m\vec{v}$

• Linear momentum is a vector quantity because it equals the product of a scalar quantity $m$ and a vector quantity $\vec{v}$.

• The SI units of momentum are kg * m / s

• Linear momentum is related to kinetic energy $\qquad K = \frac{p^2}{2m}$

Example Momentum

• A 6.0 kg bowling ball is moving at 12.0 m/s
• Therefore it's momentum is $p = mv = (6.0 , \text{kg})(12.0 , \text{m/s}) = 72 , \text{kg m/s}$

Newton's Second Law in Terms of Momentum

$\qquad \sum \vec{F} = \frac{d \vec{p}}{dt} = m\vec{a}$

• The time rate of change of the momentum of a particle is equal to the resultant force acting on the particle.
• This form is valid when $m$ is constant and also when $m$ changes.

Conservation of Momentum

• Whenever two or more particles in an isolated system interact, the total momentum of the system remains constant.
• The momentum of the system is conserved, not necessarily the momentum of the individual particles.

General Form of Momentum Conservation

$\qquad \vec{p}_{total} = \vec{p}_1 + \vec{p}_2 = \text{constant}$
$\qquad \vec{p}_{1i} + \vec{p}_{2i} = \vec{p}_{1f} + \vec{p}_{2f}$

Impulse and Momentum

Impulse

$\qquad \vec{I} \equiv \Delta \vec{p} = \vec{F} \Delta t$

• Is known as the impulse approximation
• The impulse is a vector quantity
• The magnitude of the impulse is equal to the area under the force-time curve
• The impulse can also be expressed as $\qquad \vec{I} = \overline{\vec{F}} \Delta t$ where $\overline{\vec{F}}$ is the average force

Impulse Momentum Theorem

• Impulse is the change in momentum of an object $\qquad \vec{I} = \Delta \vec{p} = \vec{p}_f - \vec{p}_i = m \vec{v}_f - m \vec{v}_i$
• Is known as the impulse-momentum theorem
• Is equivalent to Newton's Second Law

Collisions

• Collision is an event during which two particles come close to each other and interact by means of forces.
• The time interval during which the velocity changes from $\vec{v}{1i}$ to $\vec{v}{1f}$ is assumed to be short.
• There are two types of collisions
• Elastic collision - both momentum and kinetic energy are conserved
• Inelastic collision - momentum is conserved but kinetic energy is not

Perfectly Inelastic Collisions

• A perfectly inelastic collision is one in which the objects stick together after the collision.
• The objects move as one mass after the collision.
• Momentum is conserved $\qquad m_1 \vec{v}{1i} + m_2 \vec{v}{2i} = (m_1 + m_2)\vec{v}_f$

Elastic Collisions

• both momentum and kinetic energy are conserved $\qquad m_1 \vec{v}{1i} + m_2 \vec{v}{2i} = m_1 \vec{v}{1f} + m_2 \vec{v}{2f}$ $\qquad \frac{1}{2} m_1 v_{1i}^2 + \frac{1}{2} m_2 v_{2i}^2 = \frac{1}{2} m_1 v_{1f}^2 + \frac{1}{2} m_2 v_{2f}^2$

Problem-Solving Strategy Collisions

• Conceptualize , isolate the system and simplify the situation
• Categorize, Elastic or inelastic?, Isolated system?
• Analyze and set up the equations
• Finalize & Consider your results

Glancing Collisions

• For a two-dimensional collision, both the $x$ and $y$ components of momentum are conserved $\qquad m_1 v_{1ix} + m_2 v_{2ix} = m_1 v_{1fx} + m_2 v_{2fx}$ and $\qquad m_1 v_{1iy} + m_2 v_{2iy} = m_1 v_{1fy} + m_2 v_{2fy}$
• If the collision is elastic, kinetic energy is also conserved.

The Center of Mass

• The center of mass of a system is the point that moves as if all of the mass were concentrated there and all of the external forces were applied there.
• The system can be a group of particles or an extended object
• For a two-particle system: $\qquad x_{CM} \equiv \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2}$

Center of Mass, cont.

• The center of mass in three dimensions is located at $\qquad x_{CM} = \frac{1}{M} \sum_i m_i x_i, \quad y_{CM} = \frac{1}{M} \sum_i m_i y_i, \quad z_{CM} = \frac{1}{M} \sum_i m_i z_i$ where $M = \sum_i m_i$ is the total mass of the system.
• The center of mass can be located outside the object

Center of Mass, Extended Object

• The object is divided into elements of mass $\Delta m_i$ with coordinates $x_i, y_i, z_i$
• The coordinates of the center of mass are $\qquad x_{CM} = \lim_{\Delta m_i \rightarrow 0} \frac{1}{M} \sum_i x_i \Delta m_i = \frac{1}{M} \int x , dm$ $\qquad y_{CM} = \lim_{\Delta m_i \rightarrow 0} \frac{1}{M} \sum_i y_i \Delta m_i = \frac{1}{M} \int y , dm$ $\qquad z_{CM} = \lim_{\Delta m_i \rightarrow 0} \frac{1}{M} \sum_i z_i \Delta m_i = \frac{1}{M} \int z , dm$
• where $M$ is the total mass of the object

Center of Mass, Motion

• The velocity of the center of mass is $\qquad \vec{v}{CM} = \frac{d \vec{r}{CM}}{dt} = \frac{1}{M} \sum_i m_i \vec{v}_i$
• The total momentum equals the total mass multiplied by the velocity of the center of mass: $\qquad \vec{p}{total} = M \vec{v}{CM}$
• The acceleration of the center of mass is $\qquad \vec{a}{CM} = \frac{d \vec{v}{CM}}{dt} = \frac{1}{M} \sum_i m_i \vec{a}_i$

Forces and the Center of Mass

• The sum of all the forces acting on the system is equal to the total mass of the system multiplied by the acceleration of the center of mass: $\qquad \sum \vec{F}{ext} = M \vec{a}{CM}$
• The system moves as if all the mass were concentrated at the center of mass and the external force were applied at that point
• Assume the system is isolated:
  • Then $\sum \vec{F}_{ext} = 0$
  • and $\vec{a}_{CM} = 0$
  • The total momentum of the system is constant

Systems of Many Particles

• total momentum of the system as the vector sum of the momenta of all the particles: $\qquad \vec{p}_{total} = \sum_i \vec{p}_i = \sum_i m_i \vec{v}_i$
• The total momentum of the system is equal to the total mass multiplied by the velocity of the center of mass: $\qquad \vec{p}{total} = M \vec{v}{CM}$
• Newton's Second Law can be extended to a system of particles: $\qquad \sum \vec{F}{ext} = \frac{d \vec{p}{total}}{dt} = M \vec{a}_{CM}$

Conservation of Momentum for a System of Particles

• An isolated system is one for which there are no external forces
• Therefore, the derivative of the total momentum of the system is zero $\qquad \sum \vec{F}{ext} = \frac{d \vec{p}{total}}{dt} = 0$
• When no external forces are present, the total momentum of the system is constant in time: $\qquad \vec{p}_{total} = \text{constant}$

Rocket Propulsion

• Thrust on the rocket is due to the expulsion of the exhaust gases
• Rockets operate on the principle of action and reaction