mathkaveli

In our previous posts, we explored how a particle traces out an ellipse and what happens when its speed changes. But motion isn’t just about where and how fast something goes — it’s also about energy. Today, we’ll explore the kinetic and potential energy of a particle moving along an elliptical path. We'll use physics to understand the motion and simulate the energies using Octave. --- The Position and Velocity Revisited Let’s return to our position vector: r(t) = < a*cos(ωt), b*sin(ωt) > From this, the velocity is: v(t) = < -a*ω*sin(ωt), b*ω*cos(ωt) > This gives us the magnitude of velocity (speed): |v(t)| = ω \* sqrt( a²*sin²(ωt) + b²*cos²(ωt) ) --- Kinetic Energy (KE) Kinetic energy is given by the formula: KE = (1/2) \* m \* |v(t)|² So for our particle of mass m , we get: KE(t) = (1/2) \* m \* ω² \* ( a²*sin²(ωt) + b²*cos²(ωt) ) This energy varies over time as the particle speeds up and slows down along the ellipse. --- Potential Energy (PE) ...

In our last post, we explored how a particle moves along an elliptical path, using a parametric position vector: r(t) = \ That model assumed a constant angular speed, ω (omega). But real life isn’t always so smooth. What happens when ω is not constant? In this post, we’ll explore how changing speed affects the motion of a particle. We’ll go from elegant ellipses to beautiful spirals, and we’ll simulate them using **Octave**, a free programming tool for doing math and science with code. --- Motion with Changing Angular Speed In our original setup, we had: x(t) = a*cos(ωt) y(t) = b*sin(ωt) If ω is constant, the motion traces out an ellipse over and over again. But what if we let the angular speed change over time — for example: ω(t) = ω₀ + k\*t This means that as time increases, the angle grows **faster** (if k > 0) or **slower** (if k r(t) = \ where θ(t) = ∫₀ᵗ ω(s) ds = ω₀\*t + (1/2)*k*t² Now we’re dealing with **non-uniform angular motion**, and the particle no ...

Have you ever watched a satellite orbit a planet or marveled at the way planets orbit the sun? Their paths often trace out more than just perfect circles — they move in ellipses. But how do we describe such motion mathematically? And how can we measure how far an object has moved if it's not traveling in a straight line? Let’s take a look at a mathematical model for this type of motion, break it down, and even compute how far the particle travels using some Octave code (a free alternative to MATLAB). --- The Position Vector: A Mathematical Story In physics, the position of a moving particle in 2D can be described with a position vector. Here's a special one that traces out an ellipse: r(t) = < a*cos(ωt), b*sin(ωt) > This vector has: a : the horizontal radius of the ellipse b : the vertical radius ω (omega): the angular speed of the particle t : time --- What Shape Is This? Let’s prove it's an ellipse. We define: x(t) = a*cos(ωt) y(t...

Have you ever come across a second-order differential equation that looks like this? x²y″ − 3xy′ + 4y = 0 This is what's called a Cauchy-Euler equation (also known as an equidimensional equation). These are special because they match the pattern: x²y″ + axy′ + by = 0 and we can solve them with a neat trick— try solutions of the form y = xʳ . But what happens when that method gives us a repeated root ? Let's walk through it together! Step 1: Assume a Power Function Suppose we guess a solution: y = xʳ Then its derivatives are: y′ = r·x r−1 y″ = r(r−1)·x r−2 Plug these into the equation: x²y″ − 3xy′ + 4y = x²·r(r−1)x r−2 − 3x·r x r−1 + 4 x r Simplify: xʳ [r(r−1) − 3r + 4] = 0 This gives the characteristic equation: r² − 4r + 4 = 0 → (r − 2)² = 0 We have a repeated root! r = 2 Wait... Isn't That Only One Solution? Yes! The solution we get is: y₁(x) = x² But second-order differential equations need two linearly independent...

Both constructions aim to "complete" the rationals Q \mathbb{Q} , meaning to create a number system where every "approaching" sequence or gap is fully realized as a number. Feature Dedekind Cuts Cauchy Sequences Basic object A downward-closed set of rationals without a greatest element A sequence ( q n ) (q_n) of rationals that becomes arbitrarily close together Intuition A "cut" describes everything less than a real number A sequence approximates a real number Completeness The set of cuts fills in all gaps The set of limits of Cauchy sequences fills in all gaps Key property Each cut represents a real number, possibly irrational Each equivalence class of Cauchy sequences represents a real number Real number A specific subset of Q \mathbb{Q} An equivalence class of Cauchy sequences Least Upper Bound Property The union of a set of cuts is a cut Every bounded increasing sequence of Cauchy classes converges Construc...