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Most physics textbooks show you solutions. Few show you what to do when you look at a problem and have no idea how to start. That is the subject of How to Solve It by the mathematician George Pólya, a short book first published in 1945 by Princeton University Press.
Pólya was born in Budapest in 1887 and taught at ETH Zurich and then Stanford. According to the MacTutor biography, How to Solve It sold over a million copies and was translated into 17 languages. He wrote about mathematics, but the method fits physics just as well. Here are the four steps, then a JEE-style problem that uses all of them.
What are Pólya's four steps?
From a summary of How to Solve It (2nd edition, 1957), in Pólya's own questions:
Understanding the problem. "What is the unknown? What are the data? What is the condition?" Is the condition enough to find the unknown? "Draw a figure. Introduce suitable notation."
Devising a plan. "Find the connection between the data and the unknown." "Do you know a related problem?" "Look at the unknown! And try to think of a familiar problem having the same or a similar unknown."
Carrying out the plan. "Carrying out your plan of the solution, check each step."
Looking back. "Can you check the result?" "Can you derive the solution differently?" "Can you use the result, or the method, for some other problem?"
MacTutor also quotes his advice for being stuck: "If you can't solve a problem, then there is an easier problem you can solve: find it."
Worked example: a block on a rough incline
Problem. A block starts from rest and slides 5 m down a rough incline at 30° to the horizontal. The coefficient of kinetic friction is 0.2. Find its speed at the bottom. Take g = 10 m/s².
Step 1: Understand the problem
Unknown: the speed v at the bottom.
Data: distance d = 5 m, angle 30°, μ = 0.2, starts from rest, g = 10 m/s².
Condition: the block slides, so friction is kinetic and acts up the slope.
Can the condition be satisfied? The block slides only if tan 30° (about 0.58) is greater than μ. It is, so the problem is consistent.
Figure and notation: draw the incline and three forces: weight mg, normal force N, and friction f up the slope. Take x along the slope, downhill positive.
Notice the mass is not given. That is a clue, not a mistake.
Step 2: Devise a plan
Look at the unknown: a final speed after a known distance. Which familiar problems have that unknown? Constant-acceleration kinematics (v² = u² + 2as) and the work-energy theorem. Plan: find the acceleration from Newton's second law along the slope, then use kinematics.
Step 3: Carry out the plan, checking each step
Perpendicular to the slope: N = mg cos 30°.
Friction: f = μN = μmg cos 30°.
Along the slope: ma = mg sin 30° − μmg cos 30°, so a = g(sin 30° − μ cos 30°).
Numbers: a = 10 × (0.5 − 0.2 × 0.866) ≈ 3.27 m/s².
Kinematics: v² = 0 + 2 × 3.27 × 5 ≈ 32.7, so v ≈ 5.7 m/s.
Check each step. Does friction oppose the motion, so act up the slope? Yes. Did the mass cancel? Yes, which explains why it wasn't given.
Step 4: Look back
Check a limiting case. With no friction, v = √(2gh), where h = 5 sin 30° = 2.5 m, giving about 7.1 m/s. Our answer is lower, as friction should make it.
Derive it differently. Work-energy: mgh − μmg cos 30° × d = ½mv², so v² = 2(25 − 8.66) ≈ 32.7. Same answer.
Check units. Acceleration in m/s², v² in m²/s². Consistent. See dimensional analysis checks.
Use the method elsewhere. The same plan works for a block pulled up an incline, or a charged particle in a uniform field: resolve forces along the motion, then use kinematics or energy.
How to practise the four steps
Write step 1 on paper for a week: unknown, data, condition and figure, every time, even for easy problems.
When stuck in step 2, use the easier-problem trick. Remove friction, set the angle to 0° or 90°, or drop one condition. Solve that, then add the complication back.
Make step 4 non-negotiable for every wrong answer in your error log.
Our posts on drawing the diagram first and common mistakes in JEE mechanics go deeper.
Where Fermi fits
When you're stuck at step 2, a full solution is less useful than seeing why one plan was chosen. The Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer. For ideas that are easier to see than to read, the Fermi Simulation Lab has interactive simulations.
Take one problem you got wrong this week and run it through all four steps.
