Practice 2: Mathematical Routines

Overview

AP Physics C: Mechanics Practice 2: Mathematical Routines is the science practice where you do the math of physics. You derive symbolic expressions, calculate numerical answers with units, compare quantities across scenarios, and predict how values change when variables change. In short, you take known quantities and follow a logical pathway to find what you need.

This practice shows up everywhere on the exam. It is the most heavily weighted science practice on both sections, so getting comfortable with it pays off across all seven units.

What Practice 2: Mathematical Routines Means

The grouping description says it directly: conduct analyses to derive, calculate, estimate, or predict. That covers four distinct moves.

• Derive a symbolic answer from a starting principle
• Calculate or estimate a numerical answer with correct units
• Compare quantities between scenarios or at different times in one scenario
• Predict new values using how one variable depends on another

Each of these is a separate subskill, but they share the same core idea: pick a logical pathway from what you know to what you want, then follow it cleanly.

What This Practice Requires

Here is what each subskill asks you to do.

2.A Derive a symbolic expression. Start from a fundamental principle or a reference equation, then use algebra and calculus to build an expression in terms of the given symbols. No numbers until the very end, if at all. On FRQs you are often told to "begin your derivation by writing a fundamental physics principle or an equation from the reference material."

2.B Calculate or estimate an unknown quantity with units. Plug numbers into a chosen equation or chain of equations, then report a value with the right units. Estimation counts too, so you should be able to get a reasonable magnitude even without exact inputs.

2.C Compare physical quantities. Decide whether something is larger, smaller, equal, increasing, decreasing, or constant. This can be between two objects, two setups, or two moments in a single scenario.

2.D Predict new values or factors of change. Use functional dependence, like "force is proportional to the square of speed," to figure out how a quantity scales when an input changes. Often you find a factor of change such as doubled, quartered, or halved.

Skills You Need for This Practice

• Algebra fluency for rearranging and solving symbolic equations
• Calculus for derivatives and integrals, since net force, momentum change, work, and potential energy often involve them
• Unit tracking so your final answer carries correct units
• Recognizing proportionality and scaling relationships
• Choosing a fundamental principle as a starting point, such as conservation of energy, Newton's second law, or conservation of momentum
• Reading graphs to extract slopes, areas, and values

How It Shows Up on the AP Exam

Practice 2 is assessed on both the multiple-choice and free-response sections. Based on the published weightings:

FRQ 1 is the Mathematical Routines question type, which shares a name with this science practice. Derivations and calculations are front and center there, but these subskills also appear across the other free-response question types and throughout the multiple-choice section.

Practical tip: on derivation prompts, write your starting equation first. Graders look for that fundamental principle, and it anchors the rest of your work.

Examples Across the Course

These examples come from different units to show how the same four subskills repeat throughout the course.

2.A Derivation, Unit 6 Gravitation. A rock is launched at speed 2v_esc. Using energy conservation, the total energy far away equals the launch kinetic energy minus the escape kinetic energy, giving final speed sqrt(3) v_esc. You start from conservation of energy and solve symbolically.

2.A Derivation, Unit 2 Statics. A sign of mass M hangs from two wires, each at angle theta from vertical. Balancing vertical forces gives 2T cos(theta) = Mg, so T = Mg / (2 cos(theta)). The pathway is a free-body diagram plus equilibrium.

2.B Calculation, Unit 3 Energy and Friction. A block descends 5.0 m on a frictionless track, then slides on a surface with mu_k = 0.20. Setting mgh = mu_k mg d gives d = h / mu_k = 5.0 / 0.20 = 25 m. Energy in equals energy removed by friction.

2.B Calculation, Unit 4 Momentum with Calculus. Net force is F(t) = At^2 + B with A = 1 and B = 1. Change in momentum is the integral of force over time:

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Δp = ∫₀³ (t² + 1) dt = [t³/3 + t]₀³ = 9 + 3 = 12 kg·m/s

2.C Comparison, Unit 7 Oscillations. A pendulum and a spring oscillate with the same period using 1 kg spheres. Replace both with 2 kg spheres. The pendulum period does not depend on mass, so it stays the same. The spring period is 2π sqrt(m/k), so it increases with mass.

2.D Prediction, Unit 6 Angular Momentum. A star collapses to half its radius with no mass loss. Rotational inertia of a uniform sphere scales as R^2, so halving the radius makes inertia one fourth. Conservation of angular momentum means angular velocity becomes 4ω.

2.D Prediction, Unit 3 Work. Block 2 hits the floor at 2v_1, double the speed of Block 1. Work by gravity equals kinetic energy gained, and kinetic energy scales as speed squared, so the work is 4 W_1.

How to Practice Practice 2: Mathematical Routines

• For every problem, write the starting principle before touching numbers
• Do derivations fully in symbols, then substitute numbers only at the end
• Check units at the final step to catch setup errors
• Practice scaling questions by writing the proportionality, like K ∝ v^2, then comparing
• When a force or rate varies with time or position, look for an integral or derivative
• Redo a calculation problem as a derivation, and a derivation as a numerical problem, to build both 2.A and 2.B
• For comparison questions, ask what each quantity depends on and what stays fixed

Common Mistakes

• Plugging in numbers too early and losing track of the symbolic relationship
• Dropping or mismatching units in the final answer
• Forgetting to integrate when force or net force varies with time or position
• Assuming a quantity changes when it actually does not depend on the changed variable, like pendulum period and mass
• Mixing up linear scaling with squared scaling, such as treating kinetic energy as proportional to speed instead of speed squared
• Skipping the fundamental principle on FRQ derivations, which can cost setup credit
• Confusing rotational inertia scaling, which goes as R^2, with a simple linear change

Quick Review

• Practice 2 has four moves: derive (2.A), calculate or estimate (2.B), compare (2.C), predict (2.D)
• It is the highest-weighted science practice on the exam and appears in every unit
• Start derivations with a fundamental principle and keep them symbolic until the end
• Always attach correct units to calculated answers
• For comparisons and predictions, identify functional dependence first, then apply scaling
• Use calculus when force, momentum, work, or potential energy varies continuously