Unit 2: Force and Translational Dynamics

What calculus changes, and where Topic 2.9 comes in

Three places in Unit 2 need calculus, and one of them is a topic the algebra-based course does not have at all.

Topic 2.9, Resistive Forces. Essential knowledge 2.9.A.1 defines a resistive force as a velocity-dependent force in the opposite direction of an object's velocity, and gives the example $\vec{F}_r = -k\vec{v}$. Then 2.9.A.2 says the quiet part out loud: applying Newton's second law to an object upon which a resistive force is exerted results in a differential equation for velocity. Statement 2.9.A.2.i names the method, that using separation of variables the velocity can be determined by integrating over the proper limits of integration, and 2.9.A.2.ii says the acceleration or position may then be found using initial conditions and methods of calculus once a function for velocity is determined.

This cannot exist in an algebra-based course for a structural reason rather than a difficulty one: the force depends on the velocity, the velocity on the acceleration, and the acceleration on the force, and no substitution unties that loop. Statement 2.9.A.2.iii says what falls out, that the position, velocity, and acceleration as functions of time for $\vec{F}_r = -k\vec{v}$ are exponential with asymptotes determined by the initial conditions and the forces exerted on the object.

Continuous mass distributions in Topic 2.1. Physics 1 finds a centre of mass by summing over point masses. Statement 2.1.B.3 hands you the integral instead, for a nonuniform solid considered as a collection of differential masses $dm$:

Statement 2.1.B.3.i defines linear mass density as a derivative, $\lambda = \frac{d}{d\ell}m(\ell)$, and 2.1.B.3.ii says that a given mass-density function can be integrated over the length, area, or volume of a solid to get its total mass, with $M_{\text{total}} = \int \rho(r)\, dV$ as the example.

Gravitation inside matter in Topic 2.6. Learning objective 2.6.E is an entire objective on the gravitational force exerted by a uniform spherical distribution of mass. Statement 2.6.E.1 frames it as a sum over differential masses, 2.6.E.2 introduces Newton's shell theorem, and 2.6.E.3 gives the result that inside a uniform sphere the force is proportional to distance from the centre, $F_{g,\text{partial}} = -k r_{\text{partial}}$. A linear restoring force from gravity, which is a Unit 7 oscillation hiding in a Unit 2 topic.

One thing calculus does not change in Unit 2: the second law itself. Topic 2.5 has one learning objective and one equation, $\vec{a}_{\text{sys}} = \sum \vec{F}/m_{\text{sys}}$, exactly as in the algebra-based course. The derivative form $\vec{F}_{\text{net}} = d\vec{p}/dt$ appears in Unit 4, not here.

What the CED requires across Unit 2

Unit 2 of AP Physics C: Mechanics is Force and Translational Dynamics. The course and exam description weights it at 20 to 25% of the multiple-choice section, the highest band of any unit in the course, and suggests about 15 to 25 class periods, also the widest pacing range. Unit 3 follows at 15 to 25%, Unit 4 at 10 to 20%, and Units 1, 5, 6 and 7 at 10 to 15% each. Its AP Classroom Progress Check is the only one in the course with about 30 multiple-choice questions rather than about 18.

Ten topics and nineteen learning objectives, more of both than any other unit here.

Five of the nineteen objectives sit in Topic 2.6 alone, which makes gravitation the largest single topic in the unit by objective count. The CED prints the badge 1.A beside Topic 2.10's first suggested skill while giving the wording of 1.C; both codes belong to Science Practice 1, so the table above reports the badge as printed.

The CED's framing is that students are introduced to the concept of force, an interaction between two objects or systems of objects, and that this understanding is accomplished by revisiting and building upon the representations presented in Unit 1, specifically through the introduction of the free-body diagram. Its "Building the Science Practices" page names three skills for the unit, 2.A, 2.D and 3.B, and says that alongside gaining proficiency in the use of specific force equations, Unit 2 also encourages students to derive new expressions from fundamental principles to help them make predictions using functional dependence between variables.

The "Preparing for the AP Exam" note on the unit opener is unusually specific. It ties Unit 2 to question four of the free-response section, the Qualitative/Quantitative Translation, and warns that students exposed primarily to numerical problem solving often struggle with that question because it requires them to express a conceptual understanding of course content and representations.

Eleven derived equations, and none of them is on the sheet

The CED's Required Equations page states that not all equations in the framework appear on the equation sheet, that many are provided for reference and guidance or to demonstrate the final results of derivations expected of students on the exam, and that those are denoted "Derived Equations". Unit 2 is where that label does the most work.

Eleven Unit 2 equations are marked as derived, ten with the italic label and one, 2.10.A.5.iii, called a derived equation in the sentence itself. Not one of the eleven is printed on the AP Physics C: Mechanics equation sheet.

Read that column and the study plan writes itself. Every one of those results is something you are expected to be able to produce from something that is printed, so the exercise is not memorising eleven lines, it is being able to get to each of them from $\vec{a}_{\text{sys}} = \sum \vec{F}/m_{\text{sys}}$, $\lvert \vec{F}_g \rvert = Gm_1m_2/r^2$, $a_c = v^2/r$ and $T = 1/f$.

What the sheet does print for this unit is worth knowing precisely.

• $\vec{x}_{\text{cm}} = \sum m_i \vec{x}_i / \sum m_i$, $\vec{r}_{\text{cm}} = \int \vec{r}\, dm / \int dm$ and $\lambda = \frac{d}{d\ell}m(\ell)$, all three from Topic 2.1. The integral form and the density derivative are on this sheet and not on the AP Physics 1 sheet.
• $\vec{a}_{\text{sys}} = \sum \vec{F}/m_{\text{sys}} = \vec{F}_{\text{net}}/m_{\text{sys}}$, the whole of Topic 2.5.
• $\lvert \vec{F}_g \rvert = Gm_1m_2/r^2$, $\vec{F}_s = -k\Delta\vec{x}$, $T = 1/f$, and $a_c = v^2/r = r\omega^2$. The Physics 1 sheet prints $a_c = v^2/r$ without the $r\omega^2$ half.
• One friction line for both kinds of friction: $\lvert \vec{F}_f \rvert \leq \lvert \mu \vec{F}_N \rvert$. No subscripted $\mu_s$ or $\mu_k$, and no separate equality for kinetic friction, even though the framework writes 2.7.A.2 as an equality and 2.7.B.2 as an inequality.

And one absence worth naming: there is no resistive-force equation anywhere on the sheet. Topic 2.9 exists, $\vec{F}_r = -k\vec{v}$ appears in its required content at 2.9.A.1, and the sheet prints nothing for it. Neither does the sheet print any exponential solution. A resistive-force question starts from Newton's second law and separation of variables, every time.

One more structural fact: the sheet's only second-law line for this unit is $\vec{a}_{\text{sys}} = \sum \vec{F}/m_{\text{sys}}$. Where the AP Physics 1 sheet prints $\vec{F}_{\text{net}} = \Delta\vec{p}/\Delta t = m\Delta\vec{v}/\Delta t = m\vec{a}$, this sheet prints $\vec{F}_{\text{net}} = d\vec{p}/dt$ instead, and the CED files it under Unit 4.

And one place students forget to look. The Table of Information is more than the Mechanics equation table: it also carries a separate Calculus table, and that table prints $\int \frac{dx}{x+a} = \ln\lvert x + a \rvert$ along with $\int x^n\, dx = \frac{1}{n+1}x^{n+1}$ for $n \neq -1$ and the derivatives and integrals of $e^{ax}$, $\ln(ax)$, $\sin(ax)$ and $\cos(ax)$. That logarithm is precisely the integral separation of variables produces from $m\,dv/dt = F - kv$, so the mathematical step Topic 2.9 asks for is printed even though the physics is not.

Unit 2's four boundary statements, quoted whole

Unit 2 prints four boundary statements, under Topics 2.2, 2.6, 2.8 and 2.10. Topics 2.1, 2.3, 2.4, 2.5, 2.7 and 2.9 print none, which is worth noticing before you assume Topic 2.9's differential equations are fenced off somewhere. What bounds them instead is the specific form $\vec{F}_r = -k\vec{v}$ that 2.9.A.1 and 2.9.A.2.iii both name.

Topic 2.2, on free-body diagrams, is the most procedural boundary statement in the course and it is worth following to the letter:

"AP Physics C: Mechanics and AP Physics C: Electricity and Magnetism only expect students to depict the forces exerted on objects, not the force components on free-body diagrams. On the AP Physics exams, individual forces represented on a free-body diagram must be drawn as individual straight arrows, originating on the dot and pointing in the direction of the force. Individual forces that are in the same direction must be drawn side by side, not overlapping."

Three instructions in one paragraph: no components, arrows from the dot, same-direction forces side by side. Statement 2.2.B.4 adds that a coordinate system with one axis parallel to the acceleration simplifies the translation from diagram to algebra, using an inclined plane as its example. So you tilt the axes and still do not draw the components.

Topic 2.6: "AP Physics C: Mechanics does not expect students to mathematically prove or derive Newton's shell theorem." You apply it, you do not prove it. The results at 2.6.E.2.i through 2.6.E.2.iii are yours to use: zero net gravitational force inside a thin spherical shell, a shell treated as a point mass at its centre from outside, and only a partial mass contributing inside a sphere of uniform density.

Topic 2.8: "AP Physics C: Mechanics only expects students to find the effective spring constant of systems of springs that are arranged either in series or in parallel and does not expect students to find the effective spring constant of a system in which springs are arranged in both series and parallel." Pure series, yes. Pure parallel, yes. A mixed network, no.

Topic 2.10: "AP Physics C: Mechanics does not expect students to know Kepler's first or second laws of planetary motion." Only the third, and 2.10.B.1 gives it as the derived equation $T^2 = 4\pi^2 R^3/GM$ for a satellite in circular orbit around a central body.

Traps that span more than one topic

Terminal velocity is not a property of the object. Statement 2.9.A.3 defines it as the maximum speed achieved by an object moving under the influence of a constant force and a resistive force exerted in opposite directions, and says the terminal condition is reached when the net force is zero. So it depends on the driving force too. Double the driving force and the terminal speed doubles for $\vec{F}_r = -k\vec{v}$, because $v_T = F/k$ comes from setting the net force to zero rather than from any property of the falling thing.

Static friction has no formula, it has a ceiling. Statement 2.7.B.2 says static friction adopts the value and direction required to prevent an object from slipping or sliding, and the printed relation is the inequality $\lvert \vec{F}_{f,s} \rvert \leq \lvert \mu_s \vec{F}_n \rvert$. Only 2.7.B.2.ii's derived $F_{f,s,\max} = \mu_s F_N$ is an equality, and only at the point of slipping. Setting static friction equal to $\mu_s F_N$ in a problem that is not on the verge of slipping is the single most common way to get a friction answer wrong.

Centripetal acceleration is not a force and not the only acceleration. Statement 2.10.A.2 says centripetal acceleration can result from a single force, more than one force, or components of forces. Statement 2.10.A.3 defines tangential acceleration as the rate at which speed changes, and 2.10.A.4 says the net acceleration of an object moving in a circle is the vector sum of the centripetal and tangential accelerations. Uniform circular motion is the special case where the tangential part is zero, not the definition of circular motion.

If you want the algebra-based treatment of this unit

AP Physics 1 has a Unit 2 with the same title. If that is your course, the page you want is AP Physics 1 Unit 2: Force and Translational Dynamics. If you are in AP Physics C: Mechanics, this is the page.

Both courses make this their heaviest unit, which is the most useful single fact in that table. Whichever one you are taking, this is the unit where the marks are.

Topics 2.1 through 2.8 share their titles exactly, and for most of them the physics is the same. Free-body diagrams are free-body diagrams; the third law is the third law. Two of the eight genuinely differ: Topic 2.1 adds continuous mass distributions, and Topic 2.6 adds an entire learning objective on spherical distributions of mass. From 2.9 onward the numbering itself diverges, because Physics C inserts Resistive Forces and pushes Circular Motion to 2.10.

The shared material is genuinely shared, so the site's free-body diagram guide, net force guide, static versus kinetic friction guide, inclined plane walkthrough and centripetal force guide serve both courses, as do the net force calculator and friction calculator. None of them handles a velocity-dependent force, because none of them needs to for an algebra-based course. That is the line: use them up to Topic 2.8, then come back here.

How Unit 2 is assessed

The AP Physics C: Mechanics exam is 3 hours long. Section I is 42 multiple-choice questions in 85 minutes for 50% of the score. Section II is 4 free-response questions in 95 minutes for the other 50%, one of each type in a fixed order: Mathematical Routines, Translation Between Representations, Experimental Design and Analysis, and Qualitative/Quantitative Translation. A four-function, scientific, or graphing calculator is allowed on both sections.

At 20 to 25% of the multiple-choice section, Unit 2 is worth roughly nine or ten of the 42 multiple-choice questions. Its Progress Check runs about 30 multiple-choice questions and 4 free-response questions, one of each type.

Skill 2.A, deriving a symbolic expression, carries 25 to 30% of the multiple-choice section, the largest of any single skill, and Science Practice 2 as a whole carries 40 to 45% of the free-response section. The CED lists 2.A among the suggested skills for Topics 2.6, 2.7, 2.9 and 2.10, exactly the topics with derived equations to produce.

The CED's own sample questions put Unit 2 in front of you twice in the multiple-choice set and in two of the four free-response questions. Sample multiple-choice Question 6 aligns to learning objective 2.4.A and essential knowledge 2.4.A.1 with skill 2.A, and Question 7 to 2.7.B and 2.7.B.2 with skill 3.C. Sample free-response Question 2, the Translation Between Representations question worth 12 points, aligns to seven learning objectives and six of them are from this unit: 2.2.B, 2.4.A, 2.4.B, 2.5.A, 2.7.B and 2.10.A, alongside 1.4.B. Sample free-response Question 3, the Experimental Design and Analysis question, includes 2.1.B.

So Unit 2 is the unit most likely to carry a whole free-response question, and the CED's own example puts two identical race cars on a flat, horizontal road and asks for a free-body diagram, a derivation and a graph sketch.

Six optional sample instructional activities are listed, on Topics 2.1 (two of them), 2.3, 2.5, 2.7 and 2.10. None is on Topic 2.9, so the resistive-force topic has no suggested classroom activity in the CED and a course that skims it will still look complete.