Science Practice 2: Mathematical Routines

Overview

AP Physics 2 Science Practice 2: Mathematical Routines is the science practice where you do the actual math of physics. You derive symbolic expressions, calculate or estimate unknown values with units, compare quantities across scenarios, and predict how one quantity changes when another changes. In short, this practice is about choosing a logical math pathway and following it correctly to get an answer or relationship.

This practice shows up on both the multiple-choice and free-response sections, and it covers every unit from Thermodynamics to Modern Physics. If you can take a starting equation and turn it into the thing the question asks for, you are using Science Practice 2.

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What Science Practice 2: Mathematical Routines Means

The grouping description is simple: conduct analyses to derive, calculate, estimate, or predict. That breaks into four subskills.

• 2.A Derive: Build a symbolic expression from known quantities by picking and following a logical math pathway.
• 2.B Calculate or estimate: Find an unknown quantity with units from known quantities using a logical computational pathway.
• 2.C Compare: Compare physical quantities between two or more scenarios, or at different times and locations in one scenario.
• 2.D Predict: Use functional dependence between variables to predict new values or factors of change.

The common thread is that you start from a known principle or equation and move forward step by step. You are not just plugging numbers in. You are deciding which relationship applies and how to manipulate it.

What This Practice Requires

Each subskill asks for a slightly different end product.

Notice that 2.A wants algebra with no numbers in the final answer, while 2.B wants a numerical result with correct units. 2.C and 2.D both lean on relationships, but 2.C asks "which is bigger?" and 2.D asks "by what factor does it change?"

Skills You Need for This Practice

• Read which quantities are given and which one you are solving for.
• Select a starting principle or equation that connects them. On FRQs you are often told to begin from a fundamental principle or reference equation.
• Rearrange algebra cleanly, keeping track of every variable.
• Carry and check units through every step.
• Recognize proportional relationships, including squares, cubes, inverses, and exponentials.
• Set up ratios when comparing two scenarios so common terms cancel.

A useful habit for 2.C and 2.D: write the relationship as a proportionality first, like $v_{rms} \propto \sqrt{T}$, then reason about how a change in one variable affects the other.

How It Shows Up on the AP Exam

The exam has 40 multiple-choice questions and 4 free-response questions over 3 hours, and a four-function, scientific, or graphing calculator is allowed on both sections.

• Multiple-choice: All four subskills (2.A, 2.B, 2.C, 2.D) appear here. You will see symbolic answer choices for derivations, numerical answers with units for calculations, and comparison choices for 2.C and 2.D.
• Free-response: All four subskills appear here too. FRQ 1 is the Mathematical Routines question type, which shares a name with this science practice and often asks for derivations that start from a reference equation.

Across the exam, every unit can be tested with these skills, so a derivation might involve induction in Unit 12, refraction in Unit 13, or radioactive decay in Unit 15.

Examples Across the Course

These sample questions show how each subskill looks in different units.

2.D in Thermodynamics (Unit 9). The root-mean-square speed of a monatomic ideal gas at 200 K is $v_{rms}$. Later it is $2v_{rms}$. Since $v_{rms} \propto \sqrt{T}$, doubling the speed means $T$ goes up by a factor of $2^2 = 4$. New temperature is $4 \times 200 = 800$ K.

2.B in Thermodynamics (Unit 9). A 0.050 kg metal at 373 K is placed in 0.100 kg of water at 295 K, reaching equilibrium at 300 K. Using energy conservation, heat lost by metal equals heat gained by water: $m_m c_m \Delta T_m = m_w c_w \Delta T_w$ Solving gives $c_m \approx 573 \text{ J/(kg·K)}$. The pathway is choosing the calorimetry relationship, then isolating the unknown with units.

2.A in Modern Physics (Unit 15). A radioactive sample has $N_1$ nuclei at $t_1$ and $N_2$ at $t_2$. Because decay follows $N = N_0 e^{-\lambda t}$, the number remaining between two later times relates as $N_2 = N_1 e^{-\lambda(t_2 - t_1)}$. The trick is recognizing the elapsed time is $t_2 - t_1$.

2.A in Magnetism and Electromagnetism (Unit 12). A bar of resistance $R$ and length $\ell$ moves at constant speed through field $B$ along rails. Combining motional emf, Ohm's law, and force on a current-carrying wire gives an average magnetic force of $\frac{\ell^2 B^2}{R}\left(\frac{x_f - x_i}{t_f - t_i}\right)$

where $\frac{x_f - x_i}{t_f - t_i}$ is the average speed.

2.C in Electric Force, Field, and Potential (Unit 10). Given equipotential lines, you compare the field magnitudes at two points. Closer spacing of equipotentials means a stronger field, since $E$ relates to how quickly potential changes over distance. You also decide whether the field vectors point in the same or opposite directions.

2.D in Modern Physics (Unit 15). A blackbody at temperature $T_0$ has peak wavelength $\lambda_0$ and emits power $P_0$. If the peak wavelength becomes $\lambda_0/2$, the temperature doubles (peak wavelength is inversely proportional to $T$). Power scales as $T^4$, so $P$ increases by $2^4 = 16$, giving $16P_0$.

How to Practice Science Practice 2: Mathematical Routines

These are practical suggestions, not official rules.

• For derivations (2.A), always write the starting equation first, then show each algebra step. On FRQs, beginning from a reference equation is often required.
• For calculations (2.B), set up the equation symbolically, plug in numbers last, and write units on the final answer.
• For comparisons (2.C), turn the relevant equation into a proportionality and cancel anything that is the same in both cases.
• For predictions (2.D), identify the exponent on the variable that changes. A factor change of $k$ in $x$ produces a factor change of $k^n$ in a quantity that depends on $x^n$.
• Practice across units. Pull problems from circuits, optics, waves, and modern physics so you see the same four skills in new dress.
• Redo missed multiple-choice questions and identify whether the slip was choosing the wrong equation, an algebra error, or a units error.

Common Mistakes

• Plugging in numbers too early on a derivation when the answer should stay symbolic (2.A).
• Forgetting units, or mixing units like keeping a temperature in Celsius when the relationship needs kelvin (2.B).
• Treating a comparison as equal when one variable actually changed, like assuming capacitance stays the same when plate separation doubles.
• Ignoring the exponent in a proportional relationship. Doubling a variable does not always double the result. For $T^4$ behavior, doubling gives 16 times (2.D).
• Using $t_2 + t_1$ instead of $t_2 - t_1$ when a process depends on elapsed time.
• Mixing up which equipotential spacing or which scenario corresponds to the stronger quantity in a comparison.

Quick Review

• Science Practice 2 is the math engine of AP Physics 2: derive, calculate or estimate, compare, and predict.
• 2.A gives a symbolic expression. 2.B gives a number with units. 2.C compares quantities. 2.D predicts new values or factors of change.
• All four subskills appear on both multiple-choice and free-response. FRQ 1 is the Mathematical Routines question type, which uses many of these same skills.
• For comparisons and predictions, write proportionalities and watch the exponents.
• For derivations, start from a fundamental principle and show your steps. For calculations, keep units the whole way.