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

1. 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.

2. 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.

3. 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.

4. 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.

5. 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.

6. 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.

The decision tree: which approach to use?

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

Does it involve a collision or explosion?

Use momentum conservation. Compare before vs. after. Ignore the messy middle.

Does it involve heights, speeds, springs, or 'how fast at the bottom?'

Use energy conservation. Add up KE and PE at the start, set equal to KE and PE at the end. Done.

Does it ask for acceleration or detailed motion at a specific moment?

Use Newton’s 2nd law ($F = ma$). Draw the free-body diagram, add forces, divide by mass.

Is the object just sitting there or moving at constant speed?

Use equilibrium: net force = 0. All the forces have to cancel.

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)

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.

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.

Confusing mass and weight. Mass is in kilograms (kg). Weight is in newtons (N). They’re not interchangeable.

Ignoring direction. “5 m/s” and “−5 m/s” are completely different velocities. Pick a positive direction at the start and stick to it.

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.

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