AP Physics 1 · Exam guide
AP Physics 1 Mathematical Routines Explained: Derive, Calculate and Justify
The Mathematical Routines (MR) question is worth 10 points, with a suggested 20 to 25 minutes. You use mathematics to analyze a scenario: derive a relationship in symbols, calculate values, draw a helpful representation such as a free-body diagram, and then make a claim or prediction and justify it. The numbers below are made up for practice.
What the question asks
| Piece | What you do |
|---|---|
| Derive | Start from a law and get an expression in symbols for the quantity asked about. |
| Calculate | Substitute values with units and give the numerical answer. |
| Represent | Draw a free-body diagram, a sketch, or a graph that helps the analysis. |
| Claim and justify | Make a claim or prediction and justify it with a logical, step-by-step use of physics concepts. |
A worked example
A 0.50 kg ball is thrown straight up at 8.0 m/s. Ignore air resistance and use g = 10 m/s\(^2\). (a) Derive an expression for the maximum height. (b) Calculate it. (c) A student claims that a ball thrown at twice the speed reaches twice the height. Evaluate the claim.
- At the top the speed is zero, so \(0 = v_0^2 - 2gh\), which gives \(h = v_0^2 / (2g)\).
- \(h = (8.0)^2 / (2 \times 10) = 64/20 = 3.2\) m.
- With twice the speed, \(h = (16)^2 / 20 = 12.8\) m, which is four times as high.
It gives a verdict but no reasoning.
It uses the derived expression, shows the step, and checks it with numbers.
Writing a justification
- State the claim in one sentence.
- Name the principle or equation, and say why it applies here.
- Take the steps in order, so the reader can follow from the principle to the claim.
- Compare numbers when you can, as in the 3.2 m and 12.8 m above.
Common mistakes
- Putting numbers in before deriving the symbolic expression the question asks for.
- Confusing velocity with acceleration. At the top of the path the velocity is zero, but the acceleration is still g, downward.
- A justification that restates the claim.
- Leaving off units, or the direction of a vector.
Frequently asked questions
How many points is Mathematical Routines worth?
10 points, with a suggested time of 20 to 25 minutes.
Do I always have to derive in symbols?
When a part says derive or asks for an expression, yes. Graders look for the equation in symbols before numbers are substituted.
What counts as a justification?
A logical, sequential use of physics: a principle, how it applies, and how it leads to the claim.
What value of g should I use?
Use 10 m/s squared, which this course uses to match the current exam.
Where can I find the official scoring rules?
Exam formats change a little from year to year, so check the current course and exam description on the College Board site before test day.
Keep practicing
Every lesson in the AP Physics 1 course ends with practice items and an Exam focus box, and every unit has a ten-question practice set with worked answers. These free lessons also build the skills the free-response section tests:
See the AP Physics 1 course Free lessons and lesson list
More AP Physics 1 guides (4)
Start here
Free-response and writing
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