Energy on the Ellipse: Kinetic and Potential Energy in Motion

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)

Let’s now imagine that the particle is in a potential field — such as a spring system or gravitational field. For simplicity, suppose: The vertical direction corresponds to height (like gravity),  Then the potential energy is based on y(t): PE(t) = m * g * y(t) = m * g * b * sin(ωt) This is not constant — the particle has more potential energy at the top of the ellipse. ---

Total Mechanical Energy

The total energy at any moment is: E(t) = KE(t) + PE(t) If no external forces are doing work (no friction or air resistance), then energy is conserved— though it flows between kinetic and potential forms. Let’s plot these in Octave. ---

Octave Code to Visualize Energy

We’ll use: a = 5, b = 3 (ellipse size) * ω = 2π (one cycle per second) * m = 1 kg * g = 9.8 m/s² 
% Parameters
a = 5;
b = 3;
omega = 2 * pi;
m = 1;
g = 9.8;
t = linspace(0, 1, 1000);  % One full cycle

% Position
x = a * cos(omega * t);
y = b * sin(omega * t);

% Velocity components
vx = -a * omega * sin(omega * t);
vy =  b * omega * cos(omega * t);
v_squared = vx.^2 + vy.^2;

% Energies
KE = 0.5 * m * v_squared;
PE = m * g * y;
E_total = KE + PE;

% Plot energies
plot(t, KE, 'r', t, PE, 'b', t, E_total, 'k');
legend('Kinetic Energy', 'Potential Energy', 'Total Energy');
xlabel('Time (s)');
ylabel('Energy (Joules)');
title('Energy on an Elliptical Path');
---

What Does the Plot Show?

* Kinetic and potential energy oscillate in opposite phases. * When the particle is highest (maximum PE), it is slowest (minimum KE). * When it's lowest, it moves fastest. * The **total energy remains constant** — confirming conservation of energy. ---

What Does This Teach Us?

  • Motion isn’t just geometry — it’s energy flowing back and forth.
  • Even in a simple ellipse, energy can be rich and dynamic.
  • This kind of model appears in planetary motion, spring systems, and pendulums.
---

Try This Yourself:

  • Change b to be closer to a and see what happens to the energy profile.
  • Try b = 0 to turn the ellipse into a horizontal line (pure cosine motion).
  • Add friction by subtracting a small amount from KE at each time step and observe energy loss.

Comments

Post a Comment

Popular posts from this blog

What is Mathematical Fluency?

What does it really mean for students to be mathematically fluent? If you’ve been in any math PD over the past few years, you’ve likely heard the phrase everywhere. We talk about fluency as something students should develop, strengthen, and demonstrate, but it can still feel abstract when we try to describe it in observable, classroom-ready terms. This post breaks down mathematical fluency into the two simplest frames we can use as teachers: what it looks like and what it sounds like . These descriptions can guide instruction, assessment, student goal-setting, and even walkthrough conversations with colleagues or administrators. What Mathematical Fluency Looks Like In a classroom where students are developing mathematical fluency, you see students making choices about strategies rather than following steps robotically. They use representations—number lines, diagrams, tables, graphs, manipulatives, symbolic expressions—and switch between them to make sense of a problem. They move ...
Read more

Unlocking the Group: Cosets, Cayley’s Theorem, and Lagrange’s Theorem

Mathematics is full of elegant ideas that take simple definitions and build deep, powerful structures. In group theory, three such concepts—cosets, Cayley's Theorem, and Lagrange’s Theorem—form a foundational trio that unveils the internal symmetry of algebraic systems. Let’s explore what they mean and why they matter. What’s a Coset, Really? Imagine you have a group and a smaller subgroup inside it. Now, pick any element from the big group and use it to shift every element in the subgroup by multiplying on the left. The resulting set is called a left coset . You can also do this multiplication on the right, forming a right coset . These cosets aren’t usually groups themselves—but they do break the original group into equal-sized, non-overlapping pieces. Intuitively , cosets act like “shifts” of the subgroup across the whole group. Think of tiling a floor: each tile is a copy of the same shape (the subgroup) placed in different locations (cosets). Lagrange’s Theorem: Size Matt...
Read more

Verifying the Binomial Formula with Mathematical Induction

The binomial formula states: ( z 1 + z 2 ) n = ∑ k = 0 n ( n k ) z 1 n − k z 2 k (z_1 + z_2)^n = \sum_{k=0}^n \binom{n}{k} z_1^{n-k} z_2^k where ( n k ) = n ! k ! ( n − k ) ! \binom{n}{k} = \frac{n!}{k!(n-k)!} and n = 1 , 2 , … n = 1, 2, \ldots . This is a powerful formula in algebra, expressing the expansion of ( z 1 + z 2 ) n (z_1 + z_2)^n as a sum of terms involving powers of z 1 z_1 and z 2 z_2 . We’ll verify this formula using mathematical induction . Mathematical induction is a logical method to prove that a statement holds for all positive integers n n . The process involves two steps: Base Case : Verify the formula for the smallest value of n n , typically n = 1 n = 1 . Inductive Step : Assume the formula holds for some n = k n = k , and then prove it holds for n = k + 1 n = k+1 . Step 1: Base Case ( n = 1 n = 1 ) For n = 1 n = 1 , the formula becomes: ( z 1 + z 2 ) 1 = ∑ k = 0 1 ( 1 k ) z 1 1 − k z 2 k (z_1 + z_2)^1 = \sum_{k=0}^1 \binom{1}{k} z_1^{1-k} z_2^k L...
Read more