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Energy — kinetic and potential

Energy doesn't disappear — it only changes form. Send the skater down the ramp or swing the pendulum and watch potential energy turn into kinetic energy, and friction turn it into heat.

t = 0.0 s

Tracks
Edit track Drag the white points. Double-click (or double-tap) on the track to add a point, on a point — to delete it.

Settings

60 kg
none
3.0 m
45°
none
Gravity g
View

Live formulas

values update as things move
Height h—above ground
Speed v——
Swing angle θ—from vertical
Mechanical energy—Ek + Ep

Ek = ½·m·v²—

Ep = m·g·h—

Q—

E = Ek + Ep + Q—

Period of oscillation

T0 = 2π·√(l/g)—

T exact—

T measured—

What's going on here?

In our model the skater (or the pendulum bob) is a point mass. Gravity sometimes speeds it up and sometimes slows it down — and all the while the energy just flows between different forms.

Conservation of energy

As the skater rolls down, their potential energy Ep decreases and their kinetic energy Ek increases. Going uphill it's the other way around. Without friction the sum Ek + Ep (the mechanical energy) is constant — that's why in the U-ramp the skater rises on the other side to exactly the starting height.

Formulas

Ek = ½·m·v²    Ep = m·g·h

m — mass [kg], v — speed [m/s], g — gravitational acceleration [m/s²], h — height [m]. The unit of energy is the joule: 1 J = 1 kg·m²/s².

From ½·m·v² = m·g·h we get v = √(2·g·h) — the mass cancels out! A light skater and a heavy one go equally fast; only the energy values change.

Friction and heat

Friction acts against the motion and does work that turns mechanical energy into thermal energy Q — the wheels and the track warm up. Mechanical energy decreases, but the total energy Ek + Ep + Q stays constant. Energy doesn't disappear: it dissipates as heat, so the skater rises lower and lower until they stop in a valley.

A hard landing after a jump also turns some energy into heat (and sound), which is why Q can grow even with zero friction.

Where is zero?

The level from which we measure the height h can be chosen freely — on the ground, at the bottom of the U-ramp, or even above the skater. Moving the Ep = 0 line changes the values of Ep and E (they can be negative), but it doesn't change the motion or the changes in energy: ΔEp = m·g·Δh.

Loops and jumps

At the top of the loop, the track has to “press” on the skater. That requires a speed of v ≥ √(g·R) (R — the loop radius), so without friction you need to start at least 2.5·R above the bottom of a circular loop. When the speed is too low, the track's push drops to zero — the skater leaves the track, flies along a projectile path and lands back on the track.

Pendulum

The pendulum bob also turns Ep into Ek and back. For small swings the period T0 = 2π·√(l/g) depends neither on the amplitude nor on the mass. At large angles the pendulum swings more slowly — at 90° the period is about 18% longer. Air drag turns energy into heat, so the amplitude decreases; a heavier bob loses it more slowly.

Try this

  • Set friction to 0 — the “Total” bar doesn't change.
  • Switch to the Moon: the motion is slower and the energies smaller, but the skater rises to the same height.
  • On the “Loop” track, start lower and lower. Below what height does the skater fall off?
  • Move the Ep = 0 line above the skater — Ep becomes negative.
  • In the pendulum, compare the measured period with T0 for 10° and for 120°.

Keyboard

Space
start / pause
R
reset
P
back to the starting point
← →
move the skater / bob while paused (with Shift — bigger steps)
1–6
choose a track
M
switch mode: Ramp / Pendulum
S
full / slow speed
G V Z T
grid, speedometer, zero line, trail
on the scene
, . select a track point, Shift+arrows move it, + add, Delete delete, ↑ ↓ the Ep = 0 line, Esc deselect

Challenges

0 / 4

  • Loop from a low start

    On the “Loop” track (shape unchanged), place the skater lower than 4.5 m above the ground and ride through the whole loop without falling off. To do

  • All into heat

    Turn on friction on the ramp and wait until the skater stops for good. To do

  • Negative potential energy

    Turn on the zero potential energy line and move it above the skater or the bob, so that Ep drops below zero. To do

  • Slow pendulum

    Swing the pendulum so hard that the measured period is at least 20% longer than T0 = 2π·√(l/g). To do

Did you know…?

  1. Water falling from a height of 100 m is about 0.23 °C warmer at the bottom (ΔT = g·h / c). James Joule reportedly tried to measure this effect at a waterfall in the Alps during his honeymoon.
  2. A bar of chocolate (100 g, about 2.2 MJ) would be enough to lift a 60 kg person to a height of about 3.7 km — if muscles didn't waste energy. In reality about 3/4 of that energy turns into heat.
  3. A pendulum about 99.4 cm long has a period of exactly 2 s on Earth — each swing takes one second. In the 17th and 18th centuries, such a “seconds pendulum” was proposed as the standard for the unit of length.

Model: a point mass on a smooth track (a Catmull–Rom curve), integrated with the 4th-order Runge–Kutta method with an energy-balance check. In ramp mode we ignore air drag.