> ## 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.

# Problem-Solving Toolkit

> How an engineer actually attacks a physics problem from a blank page.

## The dirty secret

Most physics problems aren't hard. They just *look* hard because they show up as a wall of text and your brain doesn't know where to start.

Professional engineers don't solve problems by being smarter than you. They solve them by **following a routine.** Same routine every time. That routine is what this page is about.

If you steal nothing else from this guide, steal this method.

## The 6-step routine

<Steps>
  <Step title="Read the problem twice. Slowly.">
    First read: get a feel for what's happening.

    Second read: *picture the scene in your head.* If you can't, the problem isn't ready to be solved yet — keep reading until you can see it.
  </Step>

  <Step title="Draw it">
    A box for the object. Arrows for forces. Labels for distances and speeds. **This is non-negotiable.** Even bad drawings beat no drawings.

    Engineers draw things even when they don't need to, because the drawing is where most mistakes get caught.
  </Step>

  <Step title="List what you know and what you want">
    Two columns:

    | Known         | Want               |
    | ------------- | ------------------ |
    | mass = 2 kg   | final velocity = ? |
    | initial v = 0 |                    |
    | force = 5 N   |                    |
    | time = 3 s    |                    |

    This step is magical. Half the time, just writing this list makes the answer obvious.
  </Step>

  <Step title="Pick the recipe (equation) that connects them">
    Look at the equation cheat-sheet for this chapter. Find the one whose **left side** is what you want and whose **right side** uses only what you know.

    If no single equation works, you'll need two — one to find an intermediate value, one to finish the job.
  </Step>

  <Step title="Plug in numbers. Keep units.">
    Always carry units through the calculation. If the final units don't match what you expected (e.g., you wanted meters but got seconds), you made a mistake. Units catch errors better than any other technique.
  </Step>

  <Step title="Sanity check">
    Look at the answer and ask: *"Does this make sense in the real world?"*

    A car shouldn't accelerate at 5,000 m/s². A person shouldn't weigh 3 N. If the number is absurd, you misplaced a decimal or used the wrong equation. Catch it now, not on the exam.
  </Step>
</Steps>

## The decision tree: which approach to use?

When you don't know whether to attack a problem with forces, energy, or momentum:

<AccordionGroup>
  <Accordion title="Does it involve a collision or explosion?" icon="car-burst">
    Use **momentum conservation**. Compare before vs. after. Ignore the messy middle.
  </Accordion>

  <Accordion title="Does it involve heights, speeds, springs, or 'how fast at the bottom?'" icon="bolt">
    Use **energy conservation**. Add up KE and PE at the start, set equal to KE and PE at the end. Done.
  </Accordion>

  <Accordion title="Does it ask for acceleration or detailed motion at a specific moment?" icon="hand-fist">
    Use **Newton's 2nd law** ($F = ma$). Draw the free-body diagram, add forces, divide by mass.
  </Accordion>

  <Accordion title="Is the object just sitting there or moving at constant speed?" icon="scale-balanced">
    Use **equilibrium**: net force = 0. All the forces have to cancel.
  </Accordion>
</AccordionGroup>

In that order. Try energy and momentum first — they're shortcuts. Forces are the heavy machinery, only when you really need them.

## Common mistakes (and how to dodge them)

<Warning>
  **Mixing units.** If your speeds are in km/h but your time is in seconds, you'll get nonsense. Convert everything to SI units (meters, kilograms, seconds) before plugging in.
</Warning>

<Warning>
  **Forgetting gravity.** Anything near Earth's surface has gravity pulling down at 9.8 m/s². If you don't include it in your free-body diagram, you'll get a wrong answer.
</Warning>

<Warning>
  **Confusing mass and weight.** Mass is in kilograms (kg). Weight is in newtons (N). They're not interchangeable.
</Warning>

<Warning>
  **Ignoring direction.** "5 m/s" and "−5 m/s" are completely different velocities. Pick a positive direction at the start and stick to it.
</Warning>

## Cheat-sheet: the equations to know cold

| Topic          | Equation                                  | Plain English |
| -------------- | ----------------------------------------- | ------------- |
| Motion         | New speed = old speed + acceleration×time |               |
| Motion         | Distance traveled while accelerating      |               |
| Motion         | Speed after a distance, no time needed    |               |
| Force          | Push equals stuff-times-speedup           |               |
| Weight         | Weight = mass × gravity                   |               |
| Energy         | Energy of motion                          |               |
| Energy         | Energy stored by height                   |               |
| Work           | Force times distance                      |               |
| Power          | Energy per second                         |               |
| Momentum       | Oomph                                     |               |
| Pressure       | Force per area                            |               |
| Ohm's Law      | Voltage = current × resistance            |               |
| Electric power | Power = voltage × current                 |               |
| Waves          | Speed = frequency × wavelength            |               |

If you really get these 13 equations — not memorize, but *understand why each one says what it says* — you have the entire toolkit for high school physics and most of intro engineering.

## Final advice from an engineer

Physics doesn't reward speed. It rewards **slowing down.**

When a problem feels overwhelming, you've skipped a step. Go back. Re-read. Re-draw. Re-list. The answer always comes once the picture in your head is clear.

And remember the only thing that really matters:

> **The universe follows rules. Once you know the rules, you can predict the future.**

That's what physics is for. Welcome to the club.

<Card title="Back to: Welcome" icon="house" href="/physics/welcome">
  Revisit the table of contents.
</Card>