Energy, work, and conservation

A roller coaster is dragged up one big hill at the start, and from then on the motor is off. Every loop, dip, and rise afterward runs for free, and yet no later hill is ever taller than that first one. Something is being spent, and it is being counted exactly.

Watch a playground swing. Let go from up high and it rushes through the bottom fast, then climbs the far side and slows, stopping at almost exactly the height you released it from, before falling back. High and slow, low and fast, over and over. It looks like the swing is trading one thing for another and keeping a careful total. It is. That total is what this whole lesson is about.

First, predict

Here is the classic test. You pull a heavy pendulum bob back until it just touches your nose, hold still, and release it. It swings away, across, and comes rushing back at your face. Do you flinch?

Play with the skate park first

Before the equations, drop a skater onto the ramp and watch. Click anywhere on the track to release the skater from rest at that height, then press Play. Keep an eye on the stacked bar at the bottom: it never changes length. The colored pieces just swap places as the skater rises and falls. Then turn up the friction and watch a third color creep in and steal from the other two.

With friction at zero, the skater returns to its release height on every pass, forever, exactly like the pendulum. Turn friction up and each pass tops out a little lower, because a growing slice of the bar has gone somewhere. Nothing in the bar ever vanishes. It only moves between the three pieces.

Work: pushing something along

Start with the everyday act of doing effort on something. work is force times the distance it acts over. Shove a box across the floor with a steady push and the work you do is simply how hard, times how far.

Now watch what work does to a free object. Push a mass with a constant force over a distance, starting from rest. Using the F = ma we already have, a constant force means constant acceleration, and a little algebra on the motion gives $d=v^2/(2a)$. Substitute that into the work:

Kinetic energy: the value of motion

That result is worth a name. kinetic energy is $\frac{1}{2}m{v}^2$ (DERIVED just above, not asserted). Notice the speed is squared: go twice as fast and you carry four times the energy. This is why a crash at 60 is far more than twice as bad as one at 30, and why stopping distance balloons with speed.

Potential energy: motion held in reserve

Lift that same mass straight up instead. To raise it at a steady pace you push up with a force equal to its weight, $mg$, over a height $h$. The work you spend lifting it does not disappear; it is stored, ready to become motion the instant you let go. potential energy near the ground is $mgh$.^[2]

Conservation: the ledger that never loses a cent

Here is the payoff. As the skater drops, gravity does work on it, converting stored height energy into motion energy, cent for cent. As it climbs, the trade runs the other way. With no friction, the sum stays fixed the whole time.^[1]

That single equation explains the pendulum, the roller coaster, and the skate park at once. At the top of a swing all the energy is potential and the speed is zero; at the bottom it is all kinetic and the speed is greatest; and it can never climb higher than where it began, because there is no more in the account to spend.

The same thing in code

The skate park never tracks forces at all. It tracks the energy ledger directly: the release sets the total, height claims the potential slice, friction files away the heat slice, and whatever remains is the kinetic slice, which sets the speed. Four lines say it.

E0 = m * g * release_height     # total, set once at release (from rest)
pe = m * g * height             # claimed by current height
ke = E0 - pe - thermal          # whatever is left is motion
speed = (2 * ke / m) ** 0.5     # ...which sets the speed
# thermal only ever grows: heat is the slice that never comes back

Check yourself