Unit 1: Kinematics

What calculus changes, starting in Topic 1.2

In an algebra-based course, instantaneous velocity is the slope of a tangent line you lay a ruler against. Here it is a derivative you evaluate. Essential knowledge 1.2.C.1 states the limit in words: as the time interval used to calculate an average approaches zero, the average approaches the value of the quantity at that instant, called the instantaneous value. Statements 1.2.C.1.i and 1.2.C.1.ii then write it down.

Statement 1.2.C.2 closes it: time-dependent functions and instantaneous values of position, velocity, and acceleration can be determined using differentiation and integration. Learning objective 1.2.C asks you to describe those three quantities as a function of time, not at one labelled instant.

Topic 1.3 supplies the other direction, at 1.3.A.4.iii and 1.3.A.4.iv:

The consequence is the whole unit. Statement 1.3.A.2 opens with the words "For constant acceleration" before it prints the three kinematic equations, and that clause is doing real work. Once the acceleration varies with time, those three equations are not approximations, they are the wrong equations, and the route from acceleration to velocity to position runs through integration with an initial condition fixing each constant.

So the questions this course can ask that an algebra-based course cannot are the ones where something varies:

• a position handed to you as a polynomial in $t$ and differentiated twice,
• an acceleration given as a function of time and integrated twice,
• a velocity graph that curves, so the displacement is an integral rather than a stack of triangles and rectangles,
• a two-dimensional motion in which each component has its own nonuniform acceleration, which essential knowledge 1.5.A.2 explicitly permits when it says velocity and acceleration may be different in each dimension and may be nonuniform.

Read the derivative the other way and it locates extremes: a speed is greatest where the acceleration crosses zero. That is a calculus statement, not a graph-reading one.

What the CED requires across Unit 1

Unit 1 of AP Physics C: Mechanics is Kinematics. The course and exam description weights it at 10 to 15% of the multiple-choice section and suggests about 14 to 19 class periods. Units 5, 6 and 7 sit in the same 10 to 15% band. Unit 2 carries the top weighting at 20 to 25%, Unit 3 the widest range at 15 to 25%, and Unit 4 sits at 10 to 20%.

Five topics and eight learning objectives.

Every one of those eight objectives opens with the task verb "describe". The CED says that verb, used in nearly all learning objectives, "encompasses the range of possible graphical, mathematical, or verbal skill applications", and that students should be able to describe a physical concept graphically, mathematically, and verbally. Three of the eight sit in Topic 1.2, which is where the derivatives live.

Topics 1.4 and 1.5 print no equations at all. Relative motion and multi-dimensional motion are described in sentences in this framework rather than in formulas, which is why they carry the least to memorise and the most to get wrong.

The CED's own framing is that Unit 1 introduces students to the study of motion and serves as a foundation for all of AP Physics C: Mechanics by exploring the idea of acceleration and teaching students how representations can be used to model and analyze scientific information as it relates to the motion of objects. Its "Building the Science Practices" page names four skills for the unit, 1.A, 1.B, 2.A and 2.B, and puts the expectation this way: instead of merely evaluating equations, students need multiple opportunities to use mathematical representations to support their reasoning.

The course has one stated prerequisite. Students should have taken, or be concurrently taking, calculus. It is described as equivalent to the first course in an introductory college course sequence in calculus-based physics, and it requires that 25 percent of instructional time be spent in hands-on laboratory work.

The Unit 1 equations, and which ones the Table of Information prints

Unit 1's required content prints sixteen equation entries across its five topics, two of which repeat a derivative already given in Topic 1.2. Eight of the sixteen appear somewhere on the AP Physics C: Mechanics Table of Information, and which eight is the surprise. Note that the Table of Information is more than the Mechanics equation table: it also carries separate Vectors, Calculus, Geometry and Trigonometry, and Identities tables, and Unit 1 draws on two of those.

Four things fall out of that table.

The derivative definitions of velocity and acceleration are nowhere on the sheet. The framework writes $v_x = dx/dt$ and $a_x = dv_x/dt$ four times over, at 1.2.C.1.i, 1.2.C.1.ii, 1.3.A.4.i and 1.3.A.4.ii, and neither appears in the Mechanics table or in any other table. Both integral forms do appear. You are handed the antiderivatives and assumed to know what a derivative is.

The rotational versions are printed. The same Mechanics table prints $\omega = d\theta/dt$ and $\alpha = d\omega/dt$, which belong to Unit 5. The definitions you are given are the angular ones; the definitions you are assumed to know are the linear ones.

All of Topic 1.1's vector algebra is printed, just not in the Mechanics table. The Vectors table gives unit-vector notation with three basis vectors, the vector sum, and the component form of that sum, exactly as 1.1.A.4.i and 1.1.A.4.iii write them, plus $\vec{A} \cdot \vec{B} = AB\cos\theta$ and $\lvert \vec{A} \times \vec{B} \rvert = AB\sin\theta$ that Units 3 and 5 need. Students who only ever look at the Mechanics table miss five printed lines.

The Calculus table does more work than students expect. It prints the chain rule, the power rule for derivatives and its integral counterpart $\int x^n\, dx = \frac{1}{n+1}x^{n+1}$ for $n \neq -1$, plus derivatives and integrals of $e^{ax}$, $\ln(ax)$, $\sin(ax)$ and $\cos(ax)$, and $\int \frac{dx}{x+a} = \ln\lvert x + a\rvert$. Every integral a Unit 1 kinematics problem needs is on that list.

The printed integrals carry no limits. In the Mechanics table they read $\Delta x = \int v_x(t)\, dt$ and $\Delta v_x = \int a_x(t)\, dt$, while the framework prints both with $t_1$ and $t_2$. Elsewhere on the same table the limits are there: work is $W = \int_a^b \vec{F} \cdot d\vec{r}$ and impulse is $\vec{J} = \int_{t_1}^{t_2} \vec{F}_{\text{net}}(t)\, dt$. Supplying your own limits is your job in Unit 1, and on a free-response question it is a step worth writing down.

One more thing. The constants box holds exactly three entries: the universal gravitational constant, then $g = 9.8 \ \mathrm{m/s^2}$ as the magnitude of the acceleration due to gravity at Earth's surface, then $g = 9.8 \ \mathrm{N/kg}$ as the magnitude of the gravitational field strength there. The same page prints three exam conventions: the frame of reference of any problem is assumed to be inertial unless otherwise stated, air resistance is assumed to be negligible unless otherwise stated, and springs and strings are assumed to be ideal unless otherwise stated.

Unit 1 has three boundary statements, and one contradicts a topic title

Boundary statements are how the CED fences off a treatment. Unit 1 prints three, under Topics 1.3, 1.4 and 1.5. Topics 1.1 and 1.2 print none.

Topic 1.5, quoted whole, changes how you read the unit:

"AP Physics C: Mechanics only expects students to quantitatively analyze the motion of an object in two dimensions. AP Physics C: Electricity and Magnetism expects students to also qualitatively describe the motion of a particle in three dimensions."

The topic is called Motion in Two or Three Dimensions and the Mechanics exam will only make you calculate in two. Consistent with that, 1.1.A.4.i introduces unit-vector notation with all three basis vectors $\hat{i}$, $\hat{j}$ and $\hat{k}$, while the component sum at 1.1.A.4.iii uses only $\hat{i}$ and $\hat{j}$.

Topic 1.4 is one sentence: "Unless otherwise stated, the frame of reference of any problem may be assumed to be inertial." The exam conventions on the Table of Information page repeat it.

Topic 1.3 is the one about $g$, and its second half matters as much as its first:

"AP Physics C: Mechanics and AP Physics C: Electricity and Magnetism expects that for all situations in which a numerical quantity is required for $g$, the value $g \approx 10 \ \mathrm{m/s^2}$ will be used. However, students will not be penalized for correctly using the more precise commonly accepted values of $g = 9.81 \ \mathrm{m/s^2}$ or $g = 9.8 \ \mathrm{m/s^2}$."

Essential knowledge 1.3.A.3 agrees with the first half: near the surface of Earth the vertical acceleration caused by the force of gravity is downward, constant, and has a measured value approximately equal to $a_g = g \approx 10 \ \mathrm{m/s^2}$. And yet the equation sheet's constants table prints $g = 9.8 \ \mathrm{m/s^2}$. Both are in the same document and both are real.

The exam works in 10 to keep arithmetic out of the way, and 9.8 is explicitly not penalised. This site uses 9.8 so its numbers agree with the printed constants table, and the gap is easy to bound: a rock dropped from rest falls $\frac{1}{2}gt^2$, which after 3.0 s is 44.1 m with $g = 9.8$ and 45.0 m with $g = 10$, a difference of 2.0%. That changes no multiple-choice answer, and on a free-response question you show the substitution anyway. Which value to use, and when has a boring answer.

Traps that span more than one topic

A negative acceleration does not mean slowing down. The CED names this misconception itself. Its "Building the Science Practices" page for Unit 1 says practice with multiple representations helps dispel common misconceptions about motion, such as exclusively using negative acceleration to describe an object slowing down. An object with negative velocity and negative acceleration is speeding up. What decides it is whether $\vec{v}$ and $\vec{a}$ point the same way, not the sign of either alone.

Zero velocity does not mean zero acceleration. At the top of a vertical throw the velocity passes through zero and the acceleration is unchanged. Ask the derivative, not the value: $a_x = dv_x/dt$ is about how $v_x$ is changing, and a function can be zero at an instant where its slope is steep. Read the same statement backwards and zero acceleration marks an extreme of speed, not a stopped object.

The three kinematic equations are not general. Statement 1.3.A.2 says "For constant acceleration" before printing them. Substituting the acceleration's value at one instant into $v_x = v_{x0} + a_x t$ produces a number, and the number is wrong.

An integral gives you a change, not a value. Statement 1.3.A.4.iii gives $\Delta x$, not $x$. Two integrations need two initial conditions, and dropping one is the most common way to lose a derivation.

Perpendicular components are independent. Statement 1.5.A.3 says motion in one dimension may be changed without causing a change in a perpendicular dimension. The vertical motion of a projectile does not know what the horizontal motion is doing. What couples them is time, and only time.

Velocities depend on the observer and accelerations do not. Statement 1.4.B.2 says the observed velocity results from combining the object's velocity with the velocity of the observer's frame, and 1.4.B.2.ii says the acceleration of any object is the same as measured from all inertial reference frames. A frame moving at constant velocity changes every velocity in a problem and no acceleration, which is why Newton's second law reads the same in every inertial frame.

If you want the algebra-based treatment of this unit

Two courses have a Unit 1 called Kinematics, and they are not the same unit. If you are taking AP Physics 1, the page you want is AP Physics 1 Unit 1: Kinematics. If you are taking AP Physics C: Mechanics, this is the page.

The two units carry the same share of two different exams, which is worth saying plainly because it is easy to assume the calculus course leans harder on kinematics. It does not. It asks harder questions at the same weighting.

For the three topics whose titles are identical, the honest description is that the physics is the same and the mathematics is not. A Physics 1 student reads a velocity graph; a Physics C student is handed $v_x(t)$ and integrates it. If you arrived here from a search about reading a motion graph and you are not in a calculus-based course, the Physics 1 unit page will serve you better.

Some material serves both courses, because a kinematic equation is a kinematic equation. The site's guide to the kinematic equations, its projectile motion walkthrough and the kinematics calculator all assume constant acceleration, which is exactly the regime where the two courses agree. Use them for that half of Unit 1 and set them down when the acceleration starts changing.

How Unit 1 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.

The published skill weightings are the clearest signal about what this course rewards.

Skill 2.A is the largest single multiple-choice skill at 25 to 30%, and Practice 2 as a whole carries the largest free-response weighting at 40 to 45%. The CED lists 2.A among the suggested skills for Topics 1.1, 1.3 and 1.5. Symbolic derivation, not arithmetic, is the centre of gravity of this exam. Unit 1's Progress Check runs about 18 multiple-choice questions and 4 free-response questions.

One of the CED's fifteen sample multiple-choice questions aligns to this unit: Question 2, to learning objective 1.3.A and essential knowledge 1.3.A.4 with skill 2.C, printing a velocity-versus-time graph with six labelled points and asking for which sections the acceleration is constant and nonzero. Unit 1 objectives also appear inside two of the four sample free-response questions, 1.4.B in Question 2 and 1.3.A in Question 4, in both cases alongside objectives from later units. That is the honest shape of this unit on the exam: rarely a whole question, often part of one.

Seven optional sample instructional activities are listed for Unit 1, and they cluster on the middle topics: none on 1.1, one on 1.2, three on 1.3, one on 1.4, two on 1.5. Two of the seven name calculus outright, and the Topic 1.3 one is the closest thing in the CED to a description of what the exam asks: record position-versus-time data for a pull-back toy car as it speeds up and slows down, fit a polynomial, then use calculus to predict the car's maximum speed and its initial and final magnitudes of acceleration.