AP Physics Derivation Questions: How to Get Full Marks

Quick answer: On AP Physics free-response questions, "derive" means start from an equation on the official equation sheet and work to the answer symbolically, one line at a time. Points are awarded for naming the correct starting relationship and for each algebra step, not for the boxed result. Do not substitute numbers unless the question asks for a value.

I do not sit AP Physics. I am a Cambridge student doing IGCSE and A-Levels, my favourite subject is history and my strongest is English, so physics is the one I work hardest at. What I do spend time on is reading published scoring guidelines, and the AP ones are unusually clear about where the points sit. I also used to believe what the older version of this post said, that you should memorise every derivation word for word. Having tried it for a term, I think it is close to useless. You memorise the decision, not the algebra.

"Derive" is a different instruction from "calculate"

The task verb tells you what the scorer wants, so read it before you read the physics. Derive means begin with a fundamental relationship and finish in symbols. Calculate means numbers in, number out, with units. Determine accepts either route as long as the work is visible. Justify wants reasoning in words that points at a relationship.

The consequence is brutal. If a derivation is worth four points, the guidelines typically split them roughly like this: one for a correct starting relationship, one or two for applying it to this situation, one for the final expression in the required variables. Write only the last line and you can score one out of four with a completely correct answer.

The five-line template that fits almost every derivation

Use the same skeleton every time so you never have to invent a structure under pressure.

1. Name the principle in words: "no friction acts, so mechanical energy is conserved between these two positions".
2. Write the general equation straight off the equation sheet, unmodified.
3. Apply it to this situation. State which terms are zero and why.
4. Do the algebra, one visible line per step. No mental jumps.
5. Write the final expression using only the symbols the question listed.

Step 1 is the one students skip, and it is often the cheapest point on the page. Step 3 is where the physics actually happens.

A worked example, start to finish

Question: a block of mass m is released from rest at height h on a frictionless ramp, slides onto a frictionless horizontal surface, and compresses a spring of spring constant k. Derive an expression for the maximum compression x in terms of m, h, k and physical constants.

Line 1, in words: between release and maximum compression no friction acts, so mechanical energy is conserved. The block is momentarily at rest at both instants, so kinetic energy is zero at both.

Line 2, from the sheet: gravitational potential energy is mgh, elastic potential energy is (1/2)k x^2.

Line 3, applied: mgh = (1/2) k x^2.

Line 4, algebra: x^2 = 2mgh/k.

Line 5, answer: x = sqrt(2mgh/k).

If a later part asks for a value with m = 2.0 kg, h = 0.45 m and k = 400 N/m, then x = sqrt((2)(2.0)(9.8)(0.45)/400) = sqrt(0.0441) = 0.21 m. The numbers appear only once the question asks for them, and the symbolic result does all the work.

To apply the skeleton to a question you are stuck on, paste it into the explainer and ask for the starting relationship separately from the algebra.

Three 20-second checks that save the last point

All three together cost half a minute and catch most of the errors that lose whole questions.

Units. Newtons per metre is the same as kilograms per second squared, so 2mgh/k has units of (kg)(m/s^2)(m) divided by (kg/s^2), which is metres squared. The square root gives metres. Correct.

Limiting cases. A stiffer spring, meaning larger k, gives a smaller compression, which matches intuition. Dropping from h = 0 gives x = 0. If either check gives nonsense, you have an algebra slip.

Symbol audit. Tick off each allowed symbol in your answer. If something appears that was not on the list, such as a velocity or an angle, you left a step unfinished. If a listed symbol is missing, you probably cancelled it wrongly.

Where the points actually get lost

• Starting from a memorised result. Writing v = sqrt(2gh) straight down when asked to derive it forfeits the starting-relationship point, even though the line is true.
• Substituting numbers too early. Once numbers replace symbols the algebra points cannot be awarded, and one rounding slip contaminates everything after it.
• Leaving 9.8 inside a symbolic answer. If the question says "in terms of given quantities and physical constants", g stays as g.
• Contradicting yourself. If two equations appear and one is wrong, you can lose the point even though the correct one is there. Cross out what you do not want.
• Inventing an undefined symbol. If you introduce a speed at the bottom of the ramp, say what it is before you use it.

What to memorise, since it is not the algebra

Memorise the trigger, not the derivation. There are only about eight routes in AP Physics 1, and picking the right one is most of the battle.

• Two positions, no friction, no time asked for: energy conservation.
• A collision or explosion: momentum conservation, then compare kinetic energy separately if asked.
• Constant acceleration with time involved: kinematics, usually eliminating t between two equations.
• Connected masses: separate free-body diagrams, Newton's second law on each.
• Circular motion: set the net force toward the centre equal to mv^2/r.
• Rotation: torque equals moment of inertia times angular acceleration, or angular momentum conservation.
• Oscillation: relate restoring force to displacement, then use the period relations.
• A ramp feeding into a spring or a loop: chain energy conservation across the whole path.

Put those on flashcards, situation on the front, principle on the back. Eight cards take two minutes to review and they fix the one decision that regularly goes wrong. The physics hub covers the underlying topics if a trigger does not make sense yet.

A three-week practice plan that actually works

Week one: rewrite four past derivations using the five-line template, notes open. You are learning the layout, not testing yourself.

Week two: redo the same four closed-book, seven minutes each, then mark yourself against the official scoring guidelines line by line. Award a point only for what is literally on the page, not for what you meant. That part is uncomfortable and it is the whole point.

Week three: blank-sheet drill. Five derivations in twenty-five minutes, no notes. Anything you cannot start within thirty seconds goes back into the trigger flashcards. For a second opinion on whether a line earns its point, the grader is a fair sanity check.

Test yourself

1. A question asks you to derive the speed of a block at the bottom "in terms of m, h and physical constants", and your answer contains k. What went wrong?
2. Why does writing only the correct final formula usually score one point out of four?
3. A ball is thrown horizontally from height h and you are asked for the horizontal distance travelled. Which trigger applies, and why is energy conservation not the fastest route?

FAQ

Do I still get points if my derivation goes wrong halfway?

Usually yes. Scoring is line by line, so a correct starting relationship and a correct substitution still earn their points even if the algebra collapses afterwards. That is exactly why you write every step out rather than compressing three lines into one.

Should I write g or 9.8 in a derivation?

Write g. A symbolic derivation should contain no decimal numbers unless the question supplied one. Numbers belong only in a part that explicitly asks for a value, and there you finish with units and sensible significant figures.

Can I use an expression I derived in an earlier part?

Yes, and say so in one clause, such as "using the expression for v from part a". Carrying your own earlier result forward is normally credited even if that earlier result was wrong, as long as the method after it is sound.

Is memorising derivations word for word ever useful?

Only for the two or three that appear constantly, and even then what sticks is the shape of the argument, not the exact lines. Drilling triggers pays back far more than reciting a derivation you cannot adapt when the setup changes.

In short

Derivation questions are not a memory test, they are a bookkeeping test. Name the principle, quote the sheet equation, apply it to this situation, show every algebraic line, and finish in exactly the symbols you were given. Then spend thirty seconds on units, a limiting case and a symbol audit. Do that consistently and the points arrive whether or not you have seen the question before.