This is the method I use for most problems I come across, and is how I teach people to do their own problems. It’s how I do most modeling.

The ultimate trick is compartmentalization. Most problems have many moving parts, and it’s impossible to keep them all in your head at the same time – don’t even try! It’s the fastest way to get overwhelmed and give up on ever being able to solve it. Every problem, no matter how hard, is going to be made up of easy sub-problems.

I want you to keep in your head which “mode” you are currently in. Here are the main three modes:

1. “Physics” mode. This is where you establish which quantities you know, which quantities you want to know, and write down your equations to define the problem. Usually you will need one equation for each unknown quantity.
2. “Maths” mode. This is when you manipulate your equations using algebra to solve for the unknown quantity you want.
3. “Arithmetic” mode – this is when you actually plug in your numbers to get the unknown quantity you want.

Side note – I know it’s very tempting to plug in your numbers right into your equation (doing arithmetic before you do your maths), but it’s very important to avoid this. You should avoid plugging in numbers as long as possible for the following reasons:

1. You are doing more work than you need to do – it is easier to write “v” than it is to write “50 miles/hour”
2. You may be doing more work than you need to do – some quantities might cancel out, making your arithmetic redundant.
3. You may be doing more work than you need to do – there might be another similar problem you need to do in the future but with a few quantities changed. If you solve the problem algebraically, changing a quantity is just a matter of plugging your new quantity in. Otherwise it’s a lot more work you have to do.
4. The numbers are distracting – you should be thinking about maths and how quantities are related, not simple arithmetic. We are trying to compartmentalize.
5. Rounding errors – doing the arithmetic as you go can make errors accumulate.

Here’s a very easy example of how to use this method.

Problem: A car is traveling at 50 miles per hour. How long does it take to travel 25 miles?

Step 1: Physics mode

We know velocity (50 miles per hour) and distance (25 miles). We’ll call these v and d. We want to know time. We’ll call this t.

We know the following fundamental equation for velocity:

Now we have one equation and one unknown. We are done with physics mode and completely forget we are even doing physics.

Step 2: Maths mode

We want to solve our for t. We have one equation. Multiply both sides of the equation by t.

Divide both sides by v.

Now we have t on one side of the equation. Every quantity on the other side is known, so we know we are finished doing maths and can do the arithmetic.

Step 3: Arithmetic mode

Plug in our values for v and d.

Notice we leave our units in. This is vital. If our units work out, then we know we haven’t made any mistakes, and if we ever are given one of these quantities in different units we know how to deal with them. To get the units we want, we multiply both sides of this fraction by one hour, and cancel out the miles.

The problem is solved. It takes a half hour to travel a distance of 25 miles at 50 miles per hour.

This example was quite easy, but it demonstrates the method.

Note this method isn’t just useful for physics problems and will work for any sort of quantitative problem. Just replace “physics mode” with “chemistry mode” or “business mode”, the point is during that mode is when you establish how the quantities are related with the equations you know, and which quantities you know and which you want to know.

In part 2, I’ll do another example with a more complicated projectile motion problem.