**Case Study 1: Impulse**

**Proposed Subject Usage:** Mathematics/Physics (A/AS level), Sports Science (Degree Yr 1/2)

**AQA Physics A-Level 2009 Specification:** 11.1.7 Momentum, conservation of linear momentum, and 11.1.8 Newton's laws of motion.

**Introduction:** Impulse is a measure of the change in motion of an object, calculated as the product of the force applied and the time over which this force is applied. The impulse-momentum relationship relates an object's momentum to a force and the time over which the force acts. Impulse equals the change in momentum, and momentum is calculated as mass x velocity (p = mv). Newton's second law states that F = m x a, which can be expressed as F = d(mv)/dt. Multiplying both sides by dt yields F x dt = d(mv), or F dt = mv_final - mv_initial.

**Methods:** Video footage of a sprinter leaving the blocks was digitized and calibrated, and still images were captured to demonstrate the stages of the sprint start. Data was exported into Excel, and force, momentum, and impulse were calculated using Quintic software.

**Results:**

| Variable | Front Foot | Rear Foot |
| --- | --- | --- |
| Initial Velocity (m/s) | 0.11 | 0.19 |
| Final Velocity (m/s) | 7.43 | 6.27 |
| Initial Momentum (kg*m/s) | 9.02 | 15.58 |
| Final Momentum (kg*m/s) | 609.26 | 514.14 |
| Change in Momentum (kg*m/s) | 600.24 | 498.56 |
| Impulse Time (s) | 0.36 | 0.28 |
| Maximum Force (N) | 607.84 | 515.12 |
| Maximum Acceleration (m/s^2) | 74.13 | 62.82 |
| Impulse (N*s) | 600.52 | 498.01 |

**Discussion:** The results show that the front foot produces a higher velocity, acceleration, and momentum than the rear foot, resulting in a greater impulse. The increase in acceleration from back to front helps propel the sprinter away from the blocks. A large impulse is necessary for greater acceleration from the block, which can be achieved by producing a maximal force for a long period. However, in a sprint start, reaction time is a major variable, and the greatest force possible needs to be produced quickly.

**Conclusion:** Coaches can use this method to measure impulse and improve an athlete's rate of force development, increasing their impulse from the block and overall performance. Teachers can adapt this study to compare different athletes or manipulate calculations to apply this idea to various levels of study, engaging students' interest in sporting situations and performance kinetics.


**Theorem**: Suppose $M$ is a smooth manifold and $F: M \to \R^k$ is a continuous map. Let $\delta: M \to \R$ be a positive function. Then, there exists a smooth function $\wt F: M \to \R^k$ such that $|F(x) - \wt F(x)| < \delta(x)$ for all $x \in M$. Furthermore, if $F$ is already smooth on a closed set $A$, we can choose $\wt F = F$ on $A$.

**Proof Sketch**:

1. By the extension of smooth function lemma, we can find a smooth function $F_0: M \to \R$ that agrees with $F$ on $A$. Define $U_0 = \{x \in M | | F(x) - F_0(x)| < \delta(x) \}$.
2. For each point $x \in M$, define $U_x = \{ y \in M \backslash A | |F(y) - F(x)| < \delta(x)/2 \text{ and } \delta(x)/2 < \delta(y) \}$.
3. The collection of open sets $\{U_x\}$ covers $M \backslash A$. We choose a countable subcover $\{U_{x_i}\}_{i=1}^\infty$ and set $U_i = U_{x_i}$.
4. We create a partition of unity $\{\varphi_0, \varphi_i\}$ subject to $\{U_i\}_{i=0}$. Then, we define the function $\wt F(x) = F_0(x) \varphi_0(x) + \sum_{i=1}^\infty F(x_i) \varphi_i(x)$.

**The Big Plan**: To approximate a $C^0$ map $F: N \to M$ by a $C^\infty$ map $\wt F: N \to M$ with small $C^0$ distance, we:
1. Embed $M$ into $\R^m$ for some large $m$ using $\iota: M \into \R^m$.
2. Smooth the composition $\iota \circ F: N \to M$ to get $\wt G: N \to \R^m$, which is $C^0$-close to the original image of $\iota(M)$.
3. Project $\wt G(N)$ back onto $M$ to obtain a smooth map from $N$ to $M$. This smoothing-then-project-back operation yields the desired smooth map.


Support vector machines (SVMs) are a fundamental example of the connection between convex optimization and machine learning. They are trained on labeled data and can scale well to large numbers of features or data samples. SVMs are effective in various problems, including classification and regression.

The core idea of an SVM is to find a hyperplane that separates two labeled classes in the training dataset. The goal is to maximize the distance between the two datasets and define the hyperplane by the boundary points from the classes that touch it, known as support vectors.

### Linear Classifiers

Linear classifiers use linear hyperplanes to separate two classes. The optimization problem for a linear classifier can be formulated as:

$$
\begin{align}
\underset{a,b,t}{\text{maximize}} \quad & t \\
\text{subject to}\quad & a^T x_i – b \geq t, \; \forall i = 1,\dots,M\\
& a^T y_i – b \leq -t, \; \forall i=1,\dots,N\\
& ||a||_2 \leq 1
\end{align}
$$

This problem can be solved using convex optimization techniques.

### Robust Linear Discrimination

To handle cases where the classes cannot be fully separated, slack variables can be introduced to create a soft margin. The optimization problem becomes:

$$
\begin{align}
\underset{a,b,t}{\text{minimize}} \quad & \mathbf{1}^T u + \mathbf{1}^T v \\
\text{subject to}\quad & a^T x_i – b \geq 1 – u_i, \; \forall i = 1,\dots,M\\
& a^T y_i – b \leq -1 + v_i, \; \forall i=1,\dots,N\\
& u \succeq 0, \; v \succeq 0
\end{align}
$$

### Canonical Support Vector

To reduce variance, a trade-off between the magnitude of $a$ and the penalty for misclassification can be introduced:

$$
\begin{align}
\underset{a,b,t}{\text{minimize}} \quad & ||a||_2 + \gamma \left( \mathbf{1}^T u + \mathbf{1}^T v \right) \\
\text{subject to}\quad & a^T x_i – b \geq 1 – u_i, \; \forall i = 1,\dots,M\\
& a^T y_i – b \leq -1 + v_i, \; \forall i=1,\dots,N\\
& u \succeq 0, \; v \succeq 0
\end{align}
$$

### Implementation

The optimization problems can be solved using libraries such as CvxPy. Alternatively, the Scikit-learn library provides an implementation of SVMs that can be used for classification tasks.

### Conclusion

SVMs are a powerful tool for classification and regression tasks. They can handle high-dimensional data and are robust to noise. However, they have limitations, such as being binary classifiers and requiring a good number of samples relative to the number of features. Further exploration of SVMs and their applications can be found in resources such as Stephen Boyd's book and online tutorials.


Our team is working on an interdisciplinary unit that includes metric conversions. In math class, students are learning to support their understanding of the metric system, which they are also studying in science. To begin a lesson on measuring distances, students brainstormed possible units and tools for measuring from the front to the back of the room. They discussed the difference between yardsticks and meter sticks, and then I tasked them with measuring the distance in decimeters.

The students knew what a meter was and could identify centimeters on a meter stick, but they were unsure what a decimeter was. I encouraged them to figure it out on their own, and they set about measuring the distance. When finished, they wrote their measurements on the board. It was interesting to note the different methods and results across my four classes. Some groups used two meter sticks, alternating them for more accurate measurements, while others used one meter stick and marked where it ended with a pencil or finger.

In my first class, all groups initially measured in centimeters. We reviewed the location of centimeters on the meter stick and concluded that decimeters are the multiples of 10. The students then remeasured the distance to arrive at the correct answer. With only 14 students, I could spend more time with each student as they worked through the problem.

The second class was similar to the first, but the third math class had one group that correctly figured out the decimeter measurement. The fourth class, also small, impressed me with their approach. Many students asked to use resources to help them understand what a decimeter was, consulting their science journals or math books. All groups in this class found reasonable decimeter answers, except for one that likely made a counting error. I was impressed with their initiative and understanding.


Steve Schaefer argues that a random strategy is better than a fixed one for player 1. He analyzed the game where player 1 trades when holding 7 or less and player 2 draws when holding 8 or less. However, he found that if player 1 trades an 8 with probability p, the expected outcome is (180-2p)/2197, and if player 1 holds a 7 with probability q, the expected outcome is (180-29q)/2197. Minor variations from the optimum integer solution result in a lower expected outcome for player 1.

If the rules for ties are changed, where player 2 wins all ties but must draw to break a tie if player 1 and player 2 just traded equal cards, the best integer solution for player 1 is to trade 7 and below. The expected outcome is in player 2's favor, but only by 3/2197. If player 1 trades eights as well, player 2 will change strategies and draw to a nine, improving their expected outcome to 5/2197.

However, it is wrong to conclude that player 1 should never trade eights. If player 1 occasionally trades eights, player 2 cannot afford to shift strategy. In fact, if player 1 trades eights five out of seven times at random, player 2's expected outcome is reduced to 1.571/2197. If player 1 executes this strategy, it doesn't matter whether player 2 draws to a nine or not.

The equation for player 2's expected outcome is 2197*y = x2*(-7+12*x1)+(1-x2)*(3-2*x1), where x1 is the probability that player 1 will trade an eight and x2 is the probability that player 2 will draw to a nine. To maximize their own outcome, each player must try to find the optimal values of x1 and x2.

By taking the partial derivatives of the equation, we get:
0 = 2197* dy/dx1 = 14*x2 - 2
0 = 2197* dy/dx2 = 14*x1 - 10
Solving these equations, we find that x1 = 5/7 and x2 = 1/7. This means that player 1 will always trade a seven or less, never trade a nine or more, and trade an eight five times out of seven at random. Player 2 will always draw to an eight or less, always keep a ten or more, and draw to a nine one time out of seven at random.

This variation on the rules gives the smallest margin for either player, and it is clear that non-integer strategies are possible and can lead to more interesting solutions. The key takeaway is that a random strategy can be more effective than a fixed one, and that players must adapt to their opponent's strategy to maximize their own outcome.