Topic 1.3: Representing Motion

What Topic 1.3 requires

One learning objective, 1.3.A: describe the position, velocity, and acceleration of an object using representations of that object's motion. Four essential-knowledge statements, and 1.3.A.4 carries four sub-statements, so eight in total.

• 1.3.A.1 Motion can be represented by motion diagrams, figures, graphs, equations, and narrative descriptions.
• 1.3.A.2 For constant acceleration, three kinematic equations can be used to describe instantaneous linear motion in one dimension. The CED prints all three, then a note: the equations are written to indicate motion in the $x$-direction, but they can be used in any single dimension as appropriate.
• 1.3.A.3 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 $a_g = g \approx 10 \ \mathrm{m/s^2}$.
• 1.3.A.4 Graphs of position, velocity, and acceleration as functions of time can be used to find the relationships between those quantities.
• 1.3.A.4.i An object's instantaneous velocity is the rate of change of the object's position, which is equal to the slope of a line tangent to a point on a graph of the object's position as a function of time. Relevant equation $v_x = dx/dt$.
• 1.3.A.4.ii An object's instantaneous acceleration is the rate of change of the object's velocity, which is equal to the slope of a line tangent to a point on a graph of the object's velocity as a function of time. Relevant equation $a_x = dv_x/dt$.
• 1.3.A.4.iii The displacement of an object during a time interval is equal to the area under the curve of a graph of the object's velocity as a function of time (i.e., the area bounded by the function and the horizontal axis for the appropriate interval). Relevant equation $\Delta x = \int_{t_1}^{t_2} v_x(t)\, dt$.
• 1.3.A.4.iv The change in velocity of an object during a time interval is equal to the area under the curve of a graph of the acceleration of the object as a function of time. Relevant equation $\Delta v_x = \int_{t_1}^{t_2} a_x(t)\, dt$.

Suggested skills: 1.C create qualitative sketches of graphs that represent features of a model or the behavior of the physical system; 2.A derive a symbolic expression from known quantities by selecting and following a logical mathematical pathway; 2.D predict new values or factors of change of physical quantities using functional dependence between variables; and 3.C justify or support a claim using evidence from experimental data, physical representations, or physical principles or laws.

Topic 1.3 prints one boundary statement, about $g$, quoted whole further down this page.

The boundary statement this course does not print

The eight essential-knowledge statements above are, in wording, AP Physics 1's. Read the two frameworks side by side and 1.3.A.1 through 1.3.A.4.iv match text for text. So the difference between the two Topic 1.3s is not in what is said. It is in what is fenced off, and in what is printed underneath.

AP Physics 1's Topic 1.3 carries two boundary statements. AP Physics C: Mechanics carries one. The one that is gone reads, in full:

"AP Physics 1 does not expect students to quantitatively analyze nonuniform acceleration. However, students will be expected to be able to qualitatively analyze, sketch appropriate graphs of, and discuss situations in which acceleration is nonuniform."

That sentence is the boundary of the algebra-based course, and removing it is the boundary of this one. An AP Physics 1 question can show you a curving velocity graph and ask which way the acceleration is trending. An AP Physics C question can show you the same graph and ask for a number.

Everything else on this page follows from that single deletion:

• A velocity-time graph that curves has a displacement you cannot get from triangles and rectangles. You integrate it.
• An acceleration given as a function of time is a legitimate starting point, and the three kinematic equations are then unavailable.
• A tangent line stops being a drawing you lay a ruler against and becomes a derivative you evaluate exactly.
• "Sketch the graph" becomes "derive the function", which is why skill 2.A sits on this topic in both courses but has far more to do here.

Nothing was added to the required knowledge to achieve that. A boundary was lifted.

The same four rules, with equations only this course prints

Statement 1.3.A.4 and its four sub-statements are where the graph-reading rules live, and both courses state them identically. What differs is underneath: AP Physics C prints a relevant equation beneath each of the four, and AP Physics 1 prints none.

The wording of 1.3.A.4.iii is worth reading twice, because its parenthesis does real work: the area under the curve means the area bounded by the function and the horizontal axis for the appropriate interval. Bounded by the horizontal axis, so area below the axis counts as negative and cancels area above it. An integral that comes out to zero over an interval says the object returned to where it started, not that it never moved.

Notice also what each integral produces. $\Delta x$ and $\Delta v_x$ are changes, not values. To get a position you need $x = x_0 + \int v_x\,dt$, and the $x_0$ has to come from somewhere in the question. The same holds one level up. Two integrations, two initial conditions.

The two derivative forms repeat 1.2.C.1.i and 1.2.C.1.ii exactly. The CED restates them in a graphical context on purpose: a slope and a derivative are the same object, seen once as geometry and once as calculus.

Deriving the three kinematic equations instead of memorising them

Statement 1.3.A.2 prints the three constant-acceleration equations and opens with three words that limit all of them: For constant acceleration.

In an algebra-based course those three arrive as given. Here they are consequences of 1.3.A.4.iii and 1.3.A.4.iv, and producing them is a graded skill: 2.A, derive a symbolic expression, carries 25 to 30% of this exam's multiple-choice section, its largest single skill weighting. The derivation also shows exactly where the constant-acceleration assumption enters, which memorising cannot:

1. Start from $\Delta v_x = \int a_x(t)\,dt$. If and only if $a_x$ is constant, it comes out of the integral: $\Delta v_x = a_x \int dt = a_x t$, giving the first equation.
2. Substitute that into $\Delta x = \int v_x(t)\,dt = \int \left( v_{x0} + a_x t \right) dt = v_{x0}t + \frac{1}{2}a_x t^2$, giving the second.
3. Eliminate $t$ between them algebraically to get the third, which is why the third contains no $t$ and is the one to reach for when time is neither given nor wanted.

Step 1 is the whole assumption. Pull a varying $a_x(t)$ out of an integral and everything after it is wrong, however careful the algebra. A worked example below runs the derivation symbolically.

The note under 1.3.A.2 occasionally decides a question: the equations are written for the $x$-direction but can be used in any single dimension as appropriate. Each applies along one axis at a time, with that axis's own components, which is the machinery Topic 1.5 puts to work.

Reading a graph that curves

Once nonuniform acceleration is on the table, graph reading changes in three specific ways.

A tangent is not a chord. On a curved position graph, the line through two points gives the average velocity between them; the line touching at one point gives the instantaneous velocity there. On a straight graph those coincide, which is why the distinction is invisible in an algebra-based course and load-bearing here.

Area under a curve is not a stack of triangles. Approximating a curved velocity graph with straight segments errs in a predictable direction: it undercounts under a curve bulging upward and overcounts under one that sags. A worked example below has the triangle estimate at 24 m against a true 32 m. Where the shape is a known function, integrate.

Concavity carries information. Concave up on a position graph means $a_x > 0$, concave down means $a_x < 0$, and an inflection point is where the acceleration changes sign and the velocity is at an extreme. That is the second-derivative test applied to a graph with metres on one axis.

Before reading any of the three graphs, settle which of three different questions is being asked: the value at an instant, the slope there, or the area up to there. A large value is not a large slope, and a large slope is not a large area.

The boundary statement about g, quoted whole

Topic 1.3's one boundary statement in AP Physics C: Mechanics reads:

"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 $g = 9.8 \ \mathrm{m/s^2}$."

Both halves ship together, because the second half is what makes the first survivable. The exam will use 10. You will not lose marks for 9.81 or 9.8, provided you use them correctly.

Three facts sit in tension here and all three are real:

So the booklet you are handed prints 9.8 while the framework says the questions are built around 10. That is a working instruction rather than a contradiction to resolve. Pick one, state it, and do not change it mid-question; the difference is about 2%, and free-response marks go on method. This site uses 9.8, including in the worked examples below. The guide is g 9.8 or 10 works through the longer version.

Statement 1.3.A.3 carries a second point that has nothing to do with the number. The acceleration is downward and constant, with no mention of the object's mass, its speed, or whether it is rising or falling. A ball thrown upward has the same $a_g$ throughout, including at the top where its velocity is zero.

AP Physics 1's version of the same boundary statement drops the two course names and otherwise reads identically.

What the equation sheet prints, and one detail in how

The kinematics block at the head of the Mechanics table on the AP Physics C: Mechanics Table of Information is exactly five lines, and every one of them belongs to this topic:

Three from 1.3.A.2 and two from 1.3.A.4.iii and 1.3.A.4.iv. What is not printed is the pair of derivative forms at 1.3.A.4.i and 1.3.A.4.ii, even though the rotational column of the same table prints $\omega = d\theta/dt$ and $\alpha = d\omega/dt$. Nor is 1.3.A.3's $a_g = g \approx 10 \ \mathrm{m/s^2}$ printed, though the constants box prints $g = 9.8 \ \mathrm{m/s^2}$.

One detail in how the two integrals are printed. The CED prints them with limits, $\int_{t_1}^{t_2}$. The Table of Information prints them without limits. Nothing changes physically, and the unlimited form is the more honest one for a reference table since the interval belongs to the question rather than to the equation. But it is a reminder that the sheet gives you the relationship and not the setup: choosing $t_1$ and $t_2$, and supplying $x_0$ once you have $\Delta x$, are yours.

The derivative and integral rules you need to evaluate these are printed too, in a separate Calculus table on the same appendix page, alongside separate Vectors, Geometry and Trigonometry, and Identities tables. The power rule for integration is printed there as $\int x^n dx = \frac{1}{n+1}x^{n+1}$, $n \neq -1$. If you search this site's transcribed sheet data for those, you will not find them: `src/data/equations.ts` transcribes the physics equations and the constants boxes only. They are in the real booklet.

Traps on this topic

Using the three kinematic equations on a nonuniform acceleration. The first three words of 1.3.A.2 are "For constant acceleration". Substituting the acceleration's value at one instant produces a number, and the number is wrong. A worked example below quantifies how wrong.

Treating a curved velocity graph as a triangle. Also below: 24 m against 32 m.

Forgetting that area below the axis is negative. Statement 1.3.A.4.iii says the area bounded by the function and the horizontal axis. An object that goes out and comes back has a displacement smaller than its distance travelled, and the integral gives the displacement. To get distance travelled, split the integral at every instant where $v_x = 0$ and add the absolute values.

Reporting $\Delta x$ when the question asked for $x$. The integral gives the change. Add the initial value.

Reading a value where a slope is wanted. A position graph can sit high above the axis at an instant where its slope, and so the velocity, is zero.

Assuming the tangent at the midpoint gives the average. True for constant acceleration and false in general.

Confusing an inflection point with a stop. It is where the acceleration changes sign and the speed is at an extreme, so the object is generally moving fastest there.

Silently changing the value of $g$ inside one question. Declare 9.8 or 10 once and keep it.

If you want the algebra-based version of this topic

Both courses have a Topic 1.3 called Representing Motion, and their required knowledge is the same eight statements in the same words. Saying so plainly is more useful than inventing a difference, because the real difference is entirely in the fencing.

The skill swap matches the content. AP Physics C trades 2.C, compare physical quantities between scenarios, for 2.D, predict new values or factors of change using functional dependence between variables, which is what you do once motion arrives as a function rather than as a graph to compare against another graph.

If you are studying for AP Physics 1, the page you want is Motion Graphs, Slopes, and Areas, and the guide to the kinematic equations serves both courses. If you are studying for AP Physics C: Mechanics, the next page is Topic 1.4, where a second boundary statement goes missing.