Projectile Motion Experiment

Projectile Motion Lab

• This lab examines projectile motion by launching a ball at an angle from a projectile launcher and measuring its range.
• The experimental results are compared to theoretical calculations.

Materials Used

• Projectile launcher
• Steel ball
• Meter stick
• Protractor
• Carbon paper
• White paper
• Tape

Procedure Steps

• Launcher is set up on a flat surface.
• Launch angle is set to 30 degrees using a protractor.
• Launcher height is measured.
• Steel ball is loaded and launched.
• Meter stick is used to measure the projectile's range.
• Steps are repeated five times, and the average range is calculated.
• Results are compared to the theoretical range.

Projectile Range Data

Theoretical Range Calculation

• $R = \frac{v_0^2 \sin(2\theta)}{g}$ is the formula used for calculating the theoretical range.
• $R$ represents the range.
• $v_0$ represents the initial velocity.
• $\theta$ represents the launch angle.
• $g$ represents the acceleration due to gravity, which is $9.8 m/s^2$.
• With an initial velocity of $5 m/s$ and a launch angle of 30 degrees, the theoretical range is calculated as 2.21 m.

Results Comparison

• The experimental average range was 2.5 meters and the theoretical range was 2.21 meters.
• The experimental range was higher than the theoretical range.

Possible Sources of Error

• Measurement errors when measuring the range
• Launcher accuracy, the ball may not have launched at exactly 30 degrees
• Air resistance, which could reduce the projectiles range

Conclusion

• The lab compared experimental and theoretical projectile ranges and the experimental range was slightly higher, errors may have caused this.

Linear Algebra

Determinants

• For $A \in M_{n}(\mathbb{K})$, the determinant of $A$, $\operatorname{det}(A)$, is a scalar.
• $\operatorname{det}(A)=\sum_{\sigma \in \mathfrak{S}{n}} \varepsilon(\sigma) \prod{i=1}^{n} a_{i, \sigma(i)}$, where $\mathfrak{S}_{n}$ is the set of permutations of ${1, \ldots, n}$, and $\varepsilon(\sigma)$ is the signature of the permutation $\sigma$.

Properties of Determinants

• The determinant is an alternating $n$-linear form of the column vectors of A.
• $\operatorname{det}(A) \neq 0$ if, and only if, A is invertible.
• $\operatorname{det}(A B)=\operatorname{det}(A) \operatorname{det}(B)$.
• $\operatorname{det}\left(A^{T}\right)=\operatorname{det}(A)$.
• If $A \in M_{n}(\mathbb{K})$ is triangular, then $\operatorname{det}(A)=\prod_{i=1}^{n} a_{i i}$.

Calculating Determinants

• Dimension 2: $\operatorname{det}(A)=a_{11} a_{22}-a_{12} a_{21}$.
• Dimension 3: Sarrus' Rule: $\operatorname{det}(A)=a_{11} a_{22} a_{33}+a_{12} a_{23} a_{31}+a_{13} a_{21} a_{32}-a_{13} a_{22} a_{31}-a_{11} a_{23} a_{32}-a_{12} a_{21} a_{33}$
• Development by row or column: $\operatorname{det}(A)=\sum_{i=1}^{n}(-1)^{i+j} a_{i j} \operatorname{det}\left(A_{i j}\right)=\sum_{j=1}^{n}(-1)^{i+j} a_{i j} \operatorname{det}\left(A_{i j}\right)$, where $A_{i j}$ is the matrix obtained by deleting the $i$-th row and the $j$-th column of $A$.

Elementary Operations and Determinants

• Swapping two rows (or columns) changes the sign of the determinant.
• Multiplying a row (or column) by a scalar multiplies the determinant by that scalar.
• Adding a multiple of one row (or column) to another does not change the determinant.

Vector Spaces

• A vector space over a field $\mathbb{K}$ is a set $E$ with two operations: Addition: $E \times E \rightarrow E, (x, y) \mapsto x+y$ and Multiplication by a scalar: $\mathbb{K} \times E \rightarrow E, (\lambda, x) \mapsto \lambda x$.
• These operations must satisfy associativity, commutativity, identity element, inverse element, and distributivity.

Vector Subspaces

• A subset $F$ of a vector space $E$ is a subspace if: $F$ is non-empty and for all $x, y \in F, x+y \in F$.
• For all $x \in F$ and all $\lambda \in \mathbb{K}, \lambda x \in F$.

Vector Families

• A family of vectors $\left(v_{1}, \ldots, v_{n}\right)$ is linearly independent if $\sum_{i=1}^{n} \lambda_{i} v_{i}=0 \Rightarrow \lambda_{1}=\cdots=\lambda_{n}=0$.
• A family of vectors $\left(v_{1}, \ldots, v_{n}\right)$ is generating if every vector of $E$ can be written as a linear combination of $v_{1}, \ldots, v_{n}$.
• A basis of $E$ is a linearly independent and generating family.

Dimension

• If $E$ admits a finite basis, then all bases of $E$ have the same number of elements.
• This number is called the dimension of $E$, denoted $\operatorname{dim}(E)$.

Sum of Vector Subspaces

• If $F$ and $G$ are vector subspaces of $E$, then $F+G={x+y \mid x \in F, y \in G}$ is a vector subspace of $E$.

Direct Sum

• If $F \cap G={0}$, then the sum $F+G$ is direct, denoted $F \oplus G$.
• In this case, every vector of $F+G$ can be written uniquely as the sum of a vector from $F$ and a vector from $G$.

Complement

• If $F \oplus G=E$, then $F$ and $G$ are complementary in $E$.

Matrix Rank

• The rank of a matrix $A$ is the dimension of the vector space spanned by its column vectors.

Confusion Matrix

Definition

• A confusion matrix is a table that summarizes the performance of a classification model.
• It displays the count of correct and incorrect predictions, categorized by class.

Components

• True Positives (TP): Cases in which the model correctly predicted the positive class and the number of correctly predicted positives.
• True Negatives (TN): Cases in which the model correctly predicted the negative class and the number of correctly predicted negatives.
• False Positives (FP): Cases in which the model incorrectly predicted the positive class (Type I Error).
• False Negatives (FN): Cases in which the model incorrectly predicted the negative class (Type II Error).

Confusion Matrix Example

Metrics Derived

• A variety of metrics can derive from a confusion matrix.
• Accuracy: [\frac{TP + TN}{TP + TN + FP + FN}]
• Precision: [\frac{TP}{TP + FP}]
• Recall (Sensitivity): [\frac{TP}{TP + FN}]
• Specificity: [\frac{TN}{TN + FP}]
• F1-Score: [\frac{2 \times Precision \times Recall}{Precision + Recall}]

Uses

• Useful for evaluating a classification model's performance.
• Used to identify the classes that the model has difficulty predicting.
• Aids in comparing the performance between different models.
• Used to adjust the model's parameters to improve its performance.

Practical Example

• Consider a model that predicts if a patient has a disease or not. A test yields the following confusion matrix.

• In this case, you can calculate the evaluation metrics and analyze the performance.
• TP = 80
• TN = 90
• FP = 10
• FN = 20

"School Sudoku Master" Contest Rules (Translated from Polish)

General Provisions

• These rules define the terms for the "School Sudoku Master" contest.
• The organizer is responsible for conducting the Contest.
• The Contest is for students in grades 4-8.
• The Contest will be held on May 24, 2024, at 9:00 AM in room 15.

Contest Goals

• To develop logical thinking and math skills
• Popularize Sudoku as a mental exercise among students
• To select the "School Sudoku Master"

Participation Rules

• Participation is voluntary.
• Students in grades 4-8 can join by registering with their math teacher by May 22, 2024.
• Registration implies agreement with these rules.

Contest Procedure

• The competition will be conducted in one round lasting 60 minutes.
• Participants will solve Sudoku puzzles of varying difficulties.
• Each participant will receive a set of Sudoku puzzles and an answer sheet.
• Participants must solve the puzzles independently with calculator, cell phones, or other electronic devices being prohibited
• After 60 minutes the participants must hand in their answer sheets.

Contest Committee

• The organizer will appoint a Contest Committee, which will be responsible for overseeing the Contest, grading the work, and selecting the winners.
• The Committee will consist of a Chairperson and two Members.
• The Committee’s decisions are final and binding.

Grading Criteria

• The ranking is based on the sum of correctly solved Sudoku puzzles in the shortest time.
• If several participants have the same score, the faster solving time will determine the higher rank.

Prizes

• Winners will receive prizes and diplomas.
• The organizer reserves the right to change the prizes.

Final Provisions

• The organizer will resolve any disputes not covered by these rules.
• The organizer reserves the right to change the rules.
• These rules are effective from the date of announcement.