Modeling Motion of Falling Objects

Lab 2: Modeling the Motion of Falling Objects

• The lab's objectives include constructing computational models for falling objects, comparing models with and without air resistance, and improving a model's accuracy using experimental data.

Introduction

• The motion of an object experiencing a changing acceleration, like a falling object with air resistance, is analyzed, using a computational model to predict its complex motion.

Part 1. Free Fall

• Ignoring air resistance, the only force on a falling object is gravity, resulting in a constant acceleration of $g = 9.8 \ m/s^2$.
• The process involves opening the freefall_start.py file in VS Code and examining the existing code.
• Before coding, identify in the code comments necessary objects/entities with their properties (variables) and the steps needing repetition in the calculation loop.
• The code models the ball's motion in free fall, updating force, velocity, and position within a loop that continues until the ball hits the ground.
• The realism of the resulting motion should be assessed.

Part 2. Adding Air Resistance

• The air resistance force is approximated as $|F_{air}| = B|v|^2$, where $v$ is velocity and $B$ depends on object size/shape; the force's direction opposes motion.
• The process starts by saving a copy of Part 1 code as falling_air_resistance.py.
• A reasonable value for a basketball's B constant is 0.01 Ns²/m², which can be adjusted.
• The calculation loop is modified to incorporate air resistance, where the net force is the gravitational and air resistance forces' vector sum.
• Adjusting the value of B is done to observe its effects on the ball's motion and to see if it reaches constant speed.
• Modify the code to plot the velocity of the ball as a function of time. At what time does the ball reach terminal speed? How does this time depend on the value of B?

Part 3. Modeling Real Data

• Experimental data enhances our computational model by adjusting parameters to match a coffee filter's motion in a video.
• To analyze real data, video coffee_filter.mp4 is downloaded from Canvas and imported into Tracker.
• Tracker is used to collect data on the position of the coffee filter as a function of time, and saved as a CSV file.
• The falling_air_resistance.py code from Part 2 is opened, and modified with the csv module to read data from the CSV file.
• Initial conditions and the value of B in the computational model are adjusted so that the predicted motion of the coffee filter closely matches the experimental data.
• Compare the value of B from Part 3 to the basketball's value in Part 2.
• Additional Challenge: You may also need to adjust the size of the time step dt to get a good match.
• Additional Challenge: Write code to find the best B value automatically by minimizing the difference between predicted and experimental motion.

Modes of Heat Transfer

Conduction

• Heat transfer occurs in a solid or stationary fluid due to a temperature gradient.

Convection

• Heat transfer occurs between a surface and a moving fluid at different temperatures.

Radiation

• Heat transfer occurs due to the emission of electromagnetic waves.

Conduction

Fourier's Law

• Rate of heat transfer is proportional to area and temperature gradient: $\dot{Q} \propto A \frac{dT}{dx}$.
• Expressed as: $\dot{Q} = -kA \frac{dT}{dx}$.
• $\dot{Q}$ is the rate of heat transfer measured in Watts (W).
• A is the area normal to the heat transfer direction, measured in square meters ($m^2$).
• $\frac{dT}{dx}$ is the temperature gradient, measured in Kelvin per meter (K/m).
• k is the thermal conductivity of the material, measured in Watts per meter-Kelvin (W/m.K).

Thermal Conductivity

• Material's ability to conduct heat, quantified by heat transfer rate per thickness, area, and temperature difference.
• High k indicates a good conductor of heat, such as metals.
• Low k indicates a poor conductor of heat, such as insulators.

Convection

Newton's Law of Cooling

• Rate of heat transfer is proportional to area and the temperature difference between a surface and fluid: $\dot{Q} \propto A(T_s - T_{\infty})$.
• Expressed as: $\dot{Q} = hA(T_s - T_{\infty})$.
• $\dot{Q}$ is the rate of heat transfer measured in Watts (W).
• A is the surface area measured in square meters ($m^2$).
• $T_s$ is the surface temperature measured in Kelvin (K).
• $T_{\infty}$ is the fluid temperature measured in Kelvin (K).
• h is the convection heat transfer coefficient, measured in Watts per square meter-Kelvin ($W/m^2.K$).

Convection Heat Transfer Coefficient

• Indicates heat transfer effectiveness between a surface and fluid, depending on fluid properties, flow, and surface geometry.

Radiation

Stefan-Boltzmann Law

• Rate of heat transfer is proportional to area and the fourth power of the surface temperature: $\dot{Q} \propto AT_s^4$.
• Expressed as: $\dot{Q} = \epsilon \sigma A T_s^4$.
• $\dot{Q}$ is the rate of heat transfer measured in Watts.
• A is the surface area measured in square meters ($m^2$).
• $T_s$ is the surface temperature measured in Kelvin (K).
• $\sigma$ is the Stefan-Boltzmann constant ($5.67 \times 10^{-8} W/m^2.K^4$).
• $\epsilon$ is the emissivity of the surface ranging from 0 to 1.

Emissivity

• Indicates the effectiveness of a surface emitting radiation compared to a blackbody.
• $\epsilon = 1$ signifies a blackbody (perfect emitter).
• $\epsilon = 0$ signifies a perfect reflector

Perusmääritelmiä

Määritelmä 1

• Matrix is a rectangular array of numbers, with real numbers as entries.
• A general $m \times n$ matrix has the form: $\qquad \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \ a_{21} & a_{22} & \cdots & a_{2n} \ \vdots & \vdots & \ddots & \vdots \ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}$

Merkintöjä

• $A = (a_{ij})$
• $a_{ij}$ is the entry in matrix $A$ located in the $i$-th row and $j$-th column.
• The size of a matrix is $m \times n$, where $m$ is the number of rows and $n$ is the number of columns.
• If $m = n$, the matrix is a square matrix.
• The diagonal of a square matrix consists of the entries $a_{11}, a_{22}, \dots, a_{nn}$.

Määritelmä 2

• Two matrices $A$ and $B$ are equal if they have the same size and corresponding entries are equal, i.e., $a_{ij} = b_{ij}$ for all $i$ and $j$.

Määritelmä 3

• A zero matrix is a matrix where all entries are zero.
• The $n \times n$ identity matrix is $\qquad I_n = \begin{bmatrix} 1 & 0 & \cdots & 0 \ 0 & 1 & \cdots & 0 \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & \cdots & 1 \end{bmatrix}$

Matriisien laskutoimituksia

Skalaarikertolasku

• Multiplying a matrix $A$ by a scalar $c$ means each entry in the matrix is multiplied by the scalar. If $A = (a_{ij})$, then $cA = (ca_{ij})$. For example: $\qquad 2\begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} = \begin{bmatrix} 2 & 4 \ 6 & 8 \end{bmatrix}$

Matriisien yhteenlasku

• Two matrices $A = (a_{ij})$ and $B = (b_{ij})$ of the same size can be added element-wise: $A + B = (a_{ij} + b_{ij})$. For example: $\qquad \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} + \begin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \ 10 & 12 \end{bmatrix}$

Optimización: Introducción

¿Qué es la optimización?

• Optimization is identifying the best solution to a problem with given constraints, referred to as the optimal solution.
• Optimization problems are found in many fields, including engineering, economics, and computer science.

Componentes de un problema de optimización

• Essential components of any optimization problem: a function, variables, and restrictions.
• The function that you are trying to minimize or maximize is know as "Función objetivo". For instance, when you're faced with the task of designing a bridge, you might aim to reduce its weight without compromising its structural capabilities.
• Variables are known as the components that you have control over when optimizing your funtion objecitve. In many situations this could be how thick you want your bridge beams to be, or how many supports your bridge needs.
• Restrictions are known as items needed to be followed when optimizing the funtion objecitve. For instance, the bridge would need to support a mininum amoint of weight and stay within budget.

Tipos de optimización

• Lineal, not lineal, integers, and mutliobjecttive examples of different tupes of optimizacion.
• **lineal optimizacion-**the constraints are linear, with a linear funtion objective.
• Noot Lineal Optimizacion- Either the constraints or the objecive function is non-linear. optimizaciónEntera- the variables have to be integers.
• Optimizacion Multiobjetivo- More than one objective function.

Ejemplo

• Minimum $𝑓(𝑥)=𝑥2$ with $𝑥≥2$ has the optimal solution being $𝑥=2$. $$\min_{x \in \mathbb{R}} f(x) = x^2$$ subject to $$x \geq 2$$
• Minimizes the function objective, meeting the constraints

Química General Informe de Laboratorio

Estequiometría: Reacciones en Disolución

• The lab report is from (16 may 2024)

Resumen

• The experiment looked at how the reaction of barium chloride and sodium sulfate creates a barium sulfate sediment. The goal was to figure out the stoichiometric connections among reactants to confirm mass conservation.

Introducción

• stoichiometry is the amount of products and reagents in chemical reactions. Reactions are common in chemistry and important in many fields.

Objetivos

• Goal was to study reaction stoichometry, identify excess reagent and limiting reagent, and verify mass conservation

Materiales y Métodos

Materiales

• Reagents used in the experiment 0.2 M sulfate and barium chloride
• equipment and consumables were used

Métodos

• Steps such as mixing, filtering, washing and drying in order to calculate mass of barium sulfate, was taken to measure mass of product.

Datos Experimentales

• Tabla 1: Volumen de Reactivos*

• Different solutions such as barium chloride and sulfate were measure in mL.

• Tabla 2: Masa del Precipitado $\text{BaSO}_4$*

• Each mass per experiment was measured with the dry weight.

Cálculos y Análisis

• The primary reaction was: the barium chloride solution reacting with the sodium solution which produces barium sulfate $$\text{BaCl}_2(ac) + \text{Na}_2\text{SO}_4(ac) \rightarrow \text{BaSO}_4(s) + 2\text{NaCl}(ac)$$

Discusión

• The experments conformed with The principle and reaction amount, and reaction proportion.

Conclusión

• The conservation of mass, barium and souldium solution, and stochiometry relation, were proved, verified and stichometry calculation.

Referencias

• Referenced Brown T.L and other scientititist.

Reglas de Inferencia

• Rules of inference allow deductive arguments via valid conclusions derived from premises, which is critical logic.

Reglas básicas de inferencia

1. Modus Ponens (MP)

• If A to B is true and a is true, B is true.

   A -> B
   A
   -------
   B
   ```
  

2. Modus Tollens (MT)

• if A to B is ture and B is false, A is. A -> B ¬B ------- ¬A

3. Silogismo Hipotético (SH)

• If A to B is true and B to C is true. A to C is true A -> B B -> C ------- A -> C

4. Silogismo Disyuntivo (SD)

• If A to B is true and A is false B is true. A v B ¬A ------- B

5. Adición (Add)

• B if we know A A ------- A v B

6. Simplificación (Simp)

• We konw Q when we know A to B A ^ B ------- A

7. Conjunción (Conj)

• A and be if we now a and we know b A B ------- A ^ B

8. Resolución (Res)

• A to B and - A to C when we know Q to C is ture.. A v B ¬A v C ------- B v C

Aplicación de las reglas de inferencia

• By using rules you can show what arugments are, you prove their points based on sets of principles

Algorithmic Game Theory - Winter Term 2023/24

Lecture 4: Potential Games

• A game is defined by players, their actions, and their utility functions: $G = (N, (A_i){i \in N}, (u_i){i \in N})$.
• N =Set of players.
• $A_i$=Set of Actions for player i.
• $u_i$Utility function of player(i).
• Definition 1*:
• potential functions of G.
• Definition 2*:
• Potential funciton G.

Characterization

• Theorem the the cycle is a potential game.

Properties

• Finite potiencials have equilibrium.

Algèbre linéaire

• An espace vectoriel follows a couple of mathematical operations:
• addition, and Multiplication by scalers following associative rule
• Une famille de vecteurs is an ordered of vectors,.
• A combinaison linéaire is a scala expression.
• The sous-espace vectoriel engendré the the amount you have is the vect.
• A group is called libre (or linearly independent) when you get 0.
• A family vectrice gereratrice will provide scalrs based on what you inject..

une espace base is a group of vects, free and produce.

• Dimension is vectice bassic.

Applications linéaires

• Is the funtion to keep adding and multiplication rules.
• Le scalrs that are scalred are vectors.
• Le vect that are in in the imahge are equal to one. rang the number the picture has.

Matrices

Une the tableau of numbers.

additin the math with echa one.

• Multipliticion.
• le producto is what can happen to columbs and rows,.

la tranpsonde the vect flipped

carre has same columns and thews.

idectity it will have.

inversibl have it in equal power/.

Valeurs propres et vecteurs propres

un propresvectur is the what you add.

le the amount you add

sous space propre in an ansemble scalred. it the the vect scalred. it it scalable it will aalw be vectrice. Chapter 4: Energy Bands

Forming Energy Bands

• An isolated atom possesses distinct energy levels.
• Each level accommodates a maximum number of electrons, as per Pauli's exclusion principle.

Interacting Atoms

• When two identical atoms approach closely, their wave functions overlap.
• Pauli's exclusion principle necessitates the splitting of energy levels to accommodate electrons.

Solids

• A large number of atoms in a solid create energy bands, each with numerous closely-spaced levels.
• Band width depends on wave function overlap. Forbidden zones exist between bands, known as band gaps.

Displaying Energy Bands

• Describes energy band structures using energy level diagrams:

Metals

• Their valence band is partially filled.
• Valence and Conduction bands have no e level gap in between.
• Electrons can move with an electric field..

Insulators

• Valence band is filled.
• High level gap of usually the level great then 4ev. Electrons can not move with an applied electricity.

Semiconductors

• At 0K their valence band is completely full.
• Small energy gap between valence and conduction bands, approximately leel in height
• Finite tempreture permits the movement of electrons. Thus with electrivitu levels from insulators.

Density of States

• Discusses how many energy spots one can take in one specific unit:

Defined density of states

• g(E) shows ushow many electron state are available by looking at the space given per unit.
• Given by the equation. $g(E) = \frac{dN}{dE} = \frac{\text{number of states between } E \text{ and } E + dE}{\text{volume } \times dE}$

Density in solid

• Gives us eith how state is found with eqaution to be related to all 3 dimension

$g(E) = \frac{1}{2\pi^2} (\frac{2m^*}{\hbar^2})^{3/2} E^{1/2}$

where $m^*$ is the effective mass of the electron.

Density in 2D

• gives us eqqaution but at 2 dimensions

$g(E) = \frac{m^*}{\pi \hbar^2}$

Density in 1D

• Equation given. $g(E) = \frac{1}{\pi} (\frac{2m^*}{\hbar^2})^{1/2} E^{-1/2}$

density in 0D

g(E) = 2 \delta(E)

Fermi-Dirac discuss what fermi means distribution function.

How distribution is in the fermi form.

• Tells ehat energy levels are expected.
• given by the formula
• $f(E) = \frac{1}{1 + e^{(E - E_F) / kT}}$ , where
• $E_F$ is the Fermi energy
• $k$ is Boltzmann's constant ($k \approx 8.617 \times 10^{-5} \text{eV/K}$)
• $T$ is the temperature in Kelvin

Calculation of electrons describe the electron concertartion

• given with an energy formula How intristinct is related to concentration.
• Semiconductors and there meanings

Describing N-type semiconductor

• When a intrinstic material is used

What A P-YTP SEMINCONDUCTUIR is

• When you use acceptot. conductivity and resistivity
• defines how it is based on semiconductor. Hall effect Describes

Física

Vectores

Suma de vectores

• The addition can follow these 2 methododys:

• METHOD Graph: Draw from the axis, copy and paste, draw again.

• A- analytic method: Follow all steps to get it correctly..

  $$\vec{A} = (Ax, Ay)$$

  $$\vec{B} = (Bx, By)$$

  $$\vec{A} + \vec{B} = (Ax + Bx, Ay + By)$$

Producto de vectores

• You get productos by using a scaler, and it will only be valid is using point or an X. $$\vec{A} \cdot \vec{B} = |A| |B| \cos(\theta)$$

Ejemplo de suma de vectores

if you have two vectors:

$$\vec{A} = (3, 2)$$

$$\vec{B} = (-1, 5)$$

the sum may be:

$$\vec{A} + \vec{B} = (3 + (-1), 2 + 5) = (2, 7)$$

Cinemática

Movimiento rectilíneo uniforme (MRU)

• Level velcoties is what are seen.
• Equation used $$x = x_0 + vt$$

Movimiento rectilíneo uniformemente variado (MRUV)

• Constant Acceleartion can be found.
• Equations used are

$$v = v_0 + at$$

$$x = x_0 + v_0t + \frac{1}{2}at^2$$

$$v^2 = v_0^2 + 2a(x - x_0)$$

Tiro parabólico

• Parabolic movements is what is seen.
• Equation used are

MRU

$$x = x_0 + v_{0x}t$$

MRUV

$$y = y_0 + v_{0y}t - \frac{1}{2}gt^2$$

$$v_y = v_{0y} - gt$$

Where:

• $v_{0x} = v_0 \cos(\theta)$
• $v_{0y} = v_0 \sin(\theta)$
• $g$ is the acceleration due to gravity ($9.8 m/s^2$).

Dinámica

What newtowns laws are

• Each law states specific instances for what they do
• Primera ley (ley de la inercia): An object at rest, when acted will do.
• Segunda Ley force is mass times accelerartion
  $$\vec{F} = m\vec{a}$$
• Each and object has equal measure so the second law..

The Work

• The math to find when the force is at max

$$W = \vec{F} \cdot \vec{d} = |F| |d| \cos(\theta)$$

What energy can mean

Kineetc $$K = \frac{1}{2}mv^2$$

$$U = mgh$$

How the energy is conserved

• Using the force we can predict energy given with this rule $$K_i + U_i = K_f + U_f$$