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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.

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