> ## 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.

# Motion: How Things Move

> Speed, velocity, and acceleration explained so they finally make sense.

## Start with a question

You're in a car going down the highway. The speedometer says 100 km/h. Simple, right?

Now answer this: **are you moving?**

* Compared to the road? Yes, 100 km/h.
* Compared to the person in the passenger seat? **No. Not at all.** You're sitting still next to them.

That's the first secret of motion: **motion is always *relative* to something else.** There's no such thing as "just moving." You're always moving *compared to* something.

Keep that in your pocket. It explains a lot.

## Speed vs. velocity (the real difference)

School tells you these are different and then never explains why. Here's why.

<CardGroup cols={2}>
  <Card title="Speed" icon="gauge-high">
    **How fast.** Just a number.

    *"I'm going 60 km/h."*
  </Card>

  <Card title="Velocity" icon="location-arrow">
    **How fast *and which way*.** A number plus a direction.

    *"I'm going 60 km/h north."*
  </Card>
</CardGroup>

### Why direction matters

Imagine two cars, both doing 60 km/h, heading straight at each other. Same speed. But they're about to crash — because their **velocities are opposite.**

Now imagine two cars doing 60 km/h side by side on the highway. Same speed, *same* velocity. They'll never crash.

Direction isn't a fussy detail. **Direction is the difference between a peaceful drive and a head-on collision.** That's why engineers always track it.

<Tip>
  Anything with a direction baked in is called a **vector**. Anything that's just a number is a **scalar**. Velocity = vector. Speed = scalar. You'll see those words a lot.
</Tip>

## Acceleration: the most misunderstood word in physics

In everyday English, "accelerate" means "speed up." In physics, it means something bigger:

> **Acceleration = any change in velocity.**

That includes:

* Speeding up (pressing the gas)
* Slowing down (pressing the brake)
* Turning (changing direction even at the same speed)

Yes, **turning a corner at constant speed is acceleration.** That feels weird, but it's true — because your velocity (which includes direction) is changing.

### The feeling test

Here's a trick: **if your body feels pushed into the seat, you're accelerating.**

* Floor it from a stoplight → pushed back → acceleration.
* Slam the brakes → pushed forward → acceleration (just negative).
* Sharp left turn → pushed right → acceleration.
* Cruising on the highway → no push → **no acceleration**, even at 120 km/h.

If nothing is pushing you, your velocity isn't changing. Simple as that.

## The three equations that run everything

For an object speeding up at a steady rate, three equations cover almost every problem. Don't memorize them yet — just read what they *mean*.

<Steps>
  <Step title="Final velocity">
    $v = v_0 + at$

    *"My new speed = my old speed + (how hard I'm accelerating × how long I've been doing it)."*

    Floor a car at 3 m/s² for 5 seconds and you gained 15 m/s. Obvious when you read it slowly.
  </Step>

  <Step title="Distance traveled">
    $d = v_0 t + \tfrac{1}{2} a t^2$

    *"How far I went = how far I'd have gone at my old speed + the extra distance from speeding up."*

    The $\tfrac{1}{2} a t^2$ part is the bonus distance you get because you're getting faster as you go.
  </Step>

  <Step title="Velocity from distance">
    $v^2 = v_0^2 + 2ad$

    *"If I don't care about time, just distance, here's how fast I'll be going."*

    This one is gold for stopping-distance problems.
  </Step>
</Steps>

## A worked example you can actually feel

**Problem:** You're stopped at a red light. It turns green. You accelerate at 2 m/s². How fast are you going after 4 seconds, and how far did you travel?

**Step 1 — picture it.** A car going from 0 to *something*. Pretty boring scene. Good.

**Step 2 — list what you know.**

* Starting speed $v_0 = 0$ m/s (you were stopped)
* Acceleration $a = 2$ m/s²
* Time $t = 4$ s

**Step 3 — pick the recipe.**

We want final speed → use equation 1.

$v = 0 + (2)(4) = 8 \text{ m/s}$

That's about 29 km/h. Reasonable for 4 seconds of gentle acceleration.

We want distance → use equation 2.

$d = (0)(4) + \tfrac{1}{2}(2)(4)^2 = 16 \text{ m}$

You covered 16 meters. About 4 car-lengths. Also reasonable.

<Check>
  Notice we never "did physics." We pictured a car, listed what we knew, and picked the recipe that used those ingredients. That's the whole skill.
</Check>

## The biggest misconception

> *"If something is moving, a force must be pushing it."*

**False.** This is what Aristotle thought for 2,000 years and what your brain still secretly believes. The truth (Newton figured it out): **things in motion stay in motion** until something stops them. A hockey puck on perfect ice would slide forever.

We'll unpack that on the next page, because it's the gateway to understanding *every* force in the universe.

<Card title="Next: Forces and Newton's Laws" icon="arrow-right" href="/physics/forces">
  Why things start moving, stop moving, and push back when you push them.
</Card>