> ## Documentation Index
> Fetch the complete documentation index at: https://otzr-mintlify-3a69404b.mintlify.site/llms.txt
> Use this file to discover all available pages before exploring further.

# Energy and Work

> The universe's currency — what it is, why it never disappears, and how to use it.

## What energy actually is

Forget the textbook definition. Here's the truth:

> **Energy is the ability to make something happen.**

That's it. If something can push, lift, heat, light, move, or break something else, it has energy. If it can't, it doesn't.

Energy is measured in **joules (J)**. One joule is about the energy it takes to lift a small apple one meter into the air. Small unit. Everything else is just lots of joules.

## The one rule that runs everything: conservation of energy

> **Energy is never created and never destroyed. It only changes form.**

This is maybe the most important sentence in all of physics. Burn it in.

When you eat a sandwich, chemical energy from food turns into motion energy for your muscles, which turns into heat energy when you run. The total amount of energy never changes — it just keeps switching costumes.

<Tip>
  Whenever you're stuck on a physics problem, ask: *"Where did the energy come from, and where did it go?"* That single question solves problems that look impossible with forces alone.
</Tip>

## The two main flavors of energy

### Kinetic energy — the energy of motion

Anything that's moving has kinetic energy. The formula:

$KE = \tfrac{1}{2} m v^2$

Read it slowly: kinetic energy depends on **mass** and **speed squared**.

That **squared** is huge. It means:

* Double the speed → **4×** the energy.
* Triple the speed → **9×** the energy.

This is why a car crash at 100 km/h isn't twice as bad as 50 km/h — it's **four times** as bad. Speed is way more dangerous than people think. Engineers know this. Now you do too.

### Potential energy — stored energy

Potential energy is energy that's *waiting to happen.* A boulder on a cliff isn't doing anything right now, but if it falls, it'll do plenty. That "waiting" is potential energy.

The most common kind — **gravitational potential energy** — has a simple formula:

$PE = mgh$

* $m$ = mass
* $g$ = gravity (9.8 m/s² on Earth)
* $h$ = height above the ground (or whatever you call "zero")

Higher up = more stored energy. Heavier = more stored energy. Same intuition you already have.

Other kinds of potential energy:

* **Elastic** — stretched spring, drawn bowstring
* **Chemical** — food, fuel, batteries
* **Nuclear** — atomic nuclei

All the same idea: energy stored, ready to be released.

## Work — the bridge between force and energy

In physics, **work** has a very specific meaning:

> **Work is what happens when a force moves something.**

$W = F \cdot d$

Force times distance. If you push a box with 10 N and it slides 2 m, you did 20 J of work on it.

### The weird part

If you push against a wall as hard as you can for an hour and the wall doesn't move — **you did zero work** on the wall. Physics doesn't care that you're tired. No motion = no work.

This isn't physics being cruel. It's physics being precise: **work is specifically the energy transferred to an object by a force.** No motion means no energy was transferred to the wall.

## Watching energy transform: the roller coaster

This is the example that makes everything click.

<Steps>
  <Step title="At the top of the hill">
    The coaster is barely moving. **Lots of potential energy** (high up), **almost no kinetic energy** (slow).
  </Step>

  <Step title="Halfway down">
    It's lower, so less PE. But it's moving fast now, so more KE. **The lost PE became KE.** Total energy: same.
  </Step>

  <Step title="At the bottom">
    Lowest point → minimum PE. Fastest point → maximum KE. PE got fully converted into KE.
  </Step>

  <Step title="Climbing the next hill">
    KE turns back into PE as it slows down going up. The trade reverses.
  </Step>
</Steps>

In a perfect, frictionless world, the coaster would keep going forever, trading PE and KE. In the real world, friction and air resistance slowly bleed energy off as **heat**. Energy isn't "lost" — it just left the coaster and warmed up the rails and the air a tiny bit.

<Check>
  This is why understanding energy is so powerful: you don't have to track every force at every moment. You just compare the start and the end. *"Started with X joules of PE, ended with X joules of KE."* Done.
</Check>

## Power — how *fast* energy is used

Energy is *how much*. Power is *how fast*.

$P = \frac{W}{t}$

Measured in **watts (W)**. One watt = one joule per second.

* A 60 W lightbulb uses 60 joules every second.
* A 1500 W microwave uses 1500 joules every second.
* A human running uses around 700 W.
* A car engine puts out about 100,000 W (≈ 134 horsepower).

Same energy, faster delivery = more power. Two people can lift the same box up the same stairs, but the one who runs uses more *power*, not more *energy*.

## The mechanical engineer's secret

Whenever you can choose between solving a problem with **forces** ($F = ma$) or **energy** (conservation), **try energy first.** It's almost always shorter.

Why? Because energy doesn't care about the path. A ball that falls 10 m gains the same KE whether it fell straight down, slid down a ramp, or took a loopy roller-coaster track. Forces would force you to track every wiggle. Energy just asks: *how high did you start, how high did you end?*

<Card title="Next: Momentum" icon="arrow-right" href="/physics/momentum">
  The other thing that's always conserved — and the secret to understanding collisions.
</Card>