AP Physics C: Mechanics · Topic 1.4

Topic 1.4: Reference Frames and Relative Motion

Unit 1: Kinematics10-15% of the multiple-choice section

Topic 1.4 has identical essential knowledge in both courses and prints no equations in either. One sentence differs: AP Physics 1's boundary statement restricts relative velocity to one dimension, and AP Physics C does not print that restriction. River crossings and crosswinds are fair game here.

AP Physics: Unit 1 (topics 1.4 Reference Frames and Relative Motion). AP Physics C: Mechanics Unit 1, Topic 1.4. Two learning objectives. 1.4.A describe the reference frame of a given observer, with 1.4.A.1 (the choice of reference frame will determine the direction and magnitude of quantities measured by an observer in that reference frame). 1.4.B describe the motion of objects as measured by observers in different inertial reference frames, with 1.4.B.1 (measurements from a given reference frame may be converted to measurements from another reference frame), 1.4.B.2 (the observed velocity of an object results from the combination of the object's velocity and the velocity of the observer's reference frame), 1.4.B.2.i (combining the motion of an object and the motion of an observer in a given reference frame involves the addition or subtraction of vectors) and 1.4.B.2.ii (the acceleration of any object is the same as measured from all inertial reference frames). NO equations are printed for this topic in either course, and nothing on the Table of Information corresponds to it. ONE boundary statement in AP Physics C: Mechanics, quoted whole: 'Unless otherwise stated, the frame of reference of any problem may be assumed to be inertial.' Suggested skills 1.A, 2.B, 2.C, 3.B; AP Physics 1's Topic 1.4 suggests 1.C, 2.A, 2.B, 3.C instead. THE KEY DIFFERENCE: AP Physics 1's Topic 1.4 has identical title, identical learning objectives and identical essential-knowledge text, but its boundary statement runs to TWO paragraphs. The second, absent from Physics C, reads in full: 'Adding or subtracting vectors to find relative velocities is restricted to motion along one dimension for AP Physics 1.' Its absence is what puts two-dimensional relative velocity (river crossings, crosswinds) in scope for Physics C, with the ceiling set by the Topic 1.5 boundary statement, which says AP Physics C: Mechanics only expects students to quantitatively analyze the motion of an object in two dimensions. The inertial-frame instruction is repeated in the exam-conventions box on the C: Mechanics Table of Information appendix page, which lists exactly three 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; springs and strings are assumed to be ideal unless otherwise stated.

What Topic 1.4 requires

Two learning objectives and four essential-knowledge statements, which makes this the smallest topic in Unit 1 by required content.

1.4.A Describe the reference frame of a given observer.
1.4.A.1 The choice of reference frame will determine the direction and magnitude of quantities measured by an observer in that reference frame.
1.4.B Describe the motion of objects as measured by observers in different inertial reference frames.
1.4.B.1 Measurements from a given reference frame may be converted to measurements from another reference frame.
1.4.B.2 The observed velocity of an object results from the combination of the object's velocity and the velocity of the observer's reference frame.
1.4.B.2.i Combining the motion of an object and the motion of an observer in a given reference frame involves the addition or subtraction of vectors.
1.4.B.2.ii The acceleration of any object is the same as measured from all inertial reference frames.

Suggested skills: 1.A create diagrams, tables, charts, or schematics to represent physical situations; 2.B calculate or estimate an unknown quantity with units from known quantities; 2.C compare physical quantities between two or more scenarios or at different times and/or locations within a single scenario; and 3.B apply an appropriate law, definition, theoretical relationship, or model to make a claim.

Topic 1.4 prints no equations. Not one, in either course. Relative motion is described in sentences in this framework, which is why it carries the least to memorise in Unit 1 and a disproportionate share of the mistakes: there is no formula to fall back on if you have not set the vectors up correctly.

One boundary statement, quoted below.

The sentence AP Physics 1 prints and this course does not

All four essential-knowledge statements above are word for word AP Physics 1's, including both sub-statements. Neither course prints an equation. The two learning objectives are worded identically. On required content, the two topics are the same topic.

The difference is one sentence in the boundary statement. AP Physics 1's boundary statement has two paragraphs; AP Physics C: Mechanics prints only the first.

Both courses print: "Unless otherwise stated, the frame of reference of any problem may be assumed to be inertial."

AP Physics 1 alone prints: "Adding or subtracting vectors to find relative velocities is restricted to motion along one dimension for AP Physics 1."

That second sentence is the entire difference between the two pages, and it is a large one in practice. In AP Physics 1, statement 1.4.B.2.i says combining motions involves the addition or subtraction of vectors, and the boundary statement then confines those vectors to a line, so every relative-velocity question reduces to signed numbers. AP Physics C prints 1.4.B.2.i with no such confinement, and the Topic 1.5 boundary statement sets the ceiling: AP Physics C: Mechanics expects students to quantitatively analyze the motion of an object in two dimensions.

So the honest summary is that the physics is identical and the dimensionality is not. Nothing about frames of reference changes between the courses. What changes is that the vector addition at 1.4.B.2.i is now a real vector addition, done with components or with a triangle, and a family of questions becomes available that the algebra-based course cannot pose at all:

a boat crossing a river against a current,
an aircraft flying a heading in a crosswind,
rain falling vertically that appears slanted from a moving car,
two objects moving along perpendicular roads, and the velocity of one relative to the other.

The subscript convention, which is the whole method

Write every velocity with two subscripts, in the order of what, relative to what. $\vec{v}_{AB}$ is the velocity of A as measured in B's frame. Then a single rule does all the work:

\vec{v}_{AC} = \vec{v}_{AB} + \vec{v}_{BC}

The inner subscripts match and cancel; the outer ones survive in order. That is statement 1.4.B.2 written down, and 1.4.B.2.i is the instruction that the plus sign is a vector sum.

Two consequences make it self-checking:

Reversing the order negates the vector. $\vec{v}_{BA} = -\vec{v}_{AB}$ . If a train moves at 12 m/s relative to the ground, the ground moves at 12 m/s the other way relative to the train.
The chain extends. $\vec{v}_{AD} = \vec{v}_{AB} + \vec{v}_{BC} + \vec{v}_{CD}$ , as long as the inner subscripts keep matching.

Use it in components. Statement 1.4.B.2.i's "addition or subtraction of vectors" is the operation Topic 1.1 printed at 1.1.A.4.iii: add the $\hat{i}$ parts, add the $\hat{j}$ parts. In two dimensions there is no shortcut that avoids this, and trying to combine magnitudes directly is where the marks go.

Skill 1.A on this topic is create diagrams to represent physical situations, and it is not decoration here. A labelled vector triangle with the axes drawn is usually the difference between a sign error and a correct answer, and on the free-response section Practice 1 carries 20 to 35% of the score. AP Physics 1's version of this topic suggests 1.C, qualitative sketches of graphs, instead. The switch to 1.A tracks the switch from a number line to a plane.

Two dimensions: heading is not course

In one dimension a relative-velocity question has one answer. In two, it splits into two different questions that students routinely conflate, and telling them apart is most of the topic.

	The question	What is fixed
Heading	Which way do I point?	the object's velocity relative to the medium
Course	Which way do I actually go?	the object's velocity relative to the ground

A boat pointed straight across a river does not cross straight. A pilot who wants to fly due north in a crosswind must point somewhere other than north. The two classic sub-questions follow directly:

Point straight across. The cross-stream velocity is unaffected by the current, because perpendicular components are independent, so the crossing time is the width divided by the boat's own speed. That is the shortest crossing time, and it comes with the largest downstream drift.

Land straight across. Now the upstream component of the boat's velocity must exactly cancel the current, which uses up part of the boat's speed and leaves less for crossing. The trip takes longer, and it is impossible altogether if the boat's speed relative to the water is less than the current's.

Both are worked below with the same numbers, and the times come out 30 s and 37.5 s for the same river.

The structural point behind both is the same one Topic 1.5 states at 1.5.A.3: motion in one dimension may be changed without causing a change in a perpendicular dimension. The current is entirely along the river, so it cannot change the across-river progress. It changes only where you land.

Why every inertial observer measures the same acceleration

Statement 1.4.B.2.ii asserts it: the acceleration of any object is the same as measured from all inertial reference frames. AP Physics 1 asserts the same sentence. What this course can do is show why, in two lines, because it has derivatives.

Let frame B move with constant velocity $\vec{V}$ relative to frame C. A particle's positions in the two frames differ by the displacement between the origins, so

\vec{r}_{AC} = \vec{r}_{AB} + \vec{R}, \qquad \frac{d\vec{R}}{dt} = \vec{V} = \text{constant}

Differentiate once and you get the combination rule at 1.4.B.2:

\vec{v}_{AC} = \vec{v}_{AB} + \vec{V}

Differentiate again, and because $\vec{V}$ is constant its derivative is zero:

\vec{a}_{AC} = \vec{a}_{AB}

That is 1.4.B.2.ii, and the derivation shows exactly which assumption it rests on. Constant $\vec{V}$ is the definition of inertial. If the frame accelerates, $d\vec{V}/dt \neq 0$ and the two observers disagree about acceleration by exactly that amount, which is why the boundary statement bothers to say frames may be assumed inertial unless otherwise stated.

The physical consequence is the one that matters for the rest of the course. Newton's second law involves acceleration, and acceleration is frame-invariant among inertial frames, so the laws of mechanics come out the same for every inertial observer. A ball dropped inside a train moving at constant velocity falls straight down as seen from inside and traces a parabola as seen from the platform, and both observers measure the same downward acceleration and agree about the force causing it. A worked example below runs the numbers on that.

One quantity is invariant too and worth carrying: differences. Velocities and positions are frame-dependent, but a relative velocity between two objects is the same in every inertial frame, because the frame's velocity cancels in the subtraction.

The inertial boundary statement, and where it is repeated

Topic 1.4's boundary statement in AP Physics C: Mechanics is a single sentence:

"Unless otherwise stated, the frame of reference of any problem may be assumed to be inertial."

The same instruction appears a second time, in the exam-conventions box on the Table of Information page you are handed in the exam. That box lists three conventions, and this is the first of them: 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; springs and strings are assumed to be ideal unless otherwise stated.

Read as a working instruction, it says: do not spend time deciding whether a frame is inertial. If the question does not raise it, assume it is, apply 1.4.B.2.ii, and move on. The permission is what makes the topic tractable, since strictly speaking a frame attached to the rotating Earth is not inertial and nothing in this course wants you to worry about that.

The cases where a question does raise it are the ones where an observer is visibly accelerating: inside a braking car, on a rotating platform, in a lift starting to move. There the boundary statement's "unless otherwise stated" has been triggered, 1.4.B.2.ii no longer applies, and the two observers will disagree about the acceleration.

Traps on this topic

Adding speeds instead of velocities. In two dimensions, 30 m/s combined with 40 m/s can give anything from 10 to 70 m/s depending on the angle. This is the same error as adding vector magnitudes in Topic 1.1 and it survives because in one dimension it happens to work.

Losing the subscript order. $\vec{v}_{AB}$ and $\vec{v}_{BA}$ differ by a sign, and a question that asks for "the velocity of the car relative to the truck" is asking for one specific one of the two. Write the subscripts before writing any numbers.

Confusing heading with course. Covered above, and the single most common loss on a river-crossing question.

Assuming the shortest time and the shortest path are the same crossing. They are different headings, and one of them may be impossible.

Expecting the current to change the crossing time when the boat points straight across. It does not. Perpendicular components are independent, and the current is entirely along the river.

Applying 1.4.B.2.ii to an accelerating frame. The statement says all inertial reference frames. The derivation above shows the invariance follows from $d\vec{V}/dt = 0$ and fails without it.

Assuming a quantity that is frame-dependent must be wrong in one frame. Statement 1.4.A.1 says the choice of reference frame determines the direction and magnitude of measured quantities. Two observers reporting different velocities for one object are both right. Kinetic energy and momentum are frame-dependent for the same reason, which becomes a real issue in Unit 4.

If you want the algebra-based version of this topic

Both courses have a Topic 1.4 called Reference Frames and Relative Motion, and this is a case where the required knowledge is genuinely identical and it is more useful to say so than to manufacture a distinction.

	AP Physics 1 Topic 1.4	AP Physics C Topic 1.4
Title	Reference Frames and Relative Motion	Reference Frames and Relative Motion
Learning objectives	1.4.A, 1.4.B	1.4.A, 1.4.B
Essential knowledge	1.4.A.1, 1.4.B.1, 1.4.B.2, 1.4.B.2.i, 1.4.B.2.ii	the same five, identical text
Equations printed	none	none
Boundary statement	two paragraphs	the first paragraph only
Relative velocity	restricted to one dimension	not restricted; two dimensions per the Topic 1.5 boundary
River crossings, crosswinds	out of scope	in scope
Suggested skills	1.C, 2.A, 2.B, 3.C	1.A, 2.B, 2.C, 3.B

The skill lists move with the content: AP Physics C swaps qualitative graph sketching (1.C) for diagram creation (1.A), which is what a two-dimensional vector triangle needs, and swaps justifying a claim from evidence (3.C) for applying a model to make a claim (3.B).

If you are studying for AP Physics 1, the page you want is Relative Velocity and Reference Frames, which keeps everything on a line as its boundary statement requires. If you are studying for AP Physics C: Mechanics, stay here for the two-dimensional treatment, then read Topic 1.5, which states component independence as physics.

A boat crossing a river, pointed two different ways

A river 90 m wide flows at 1.8 m/s. A boat travels at 3.0 m/s relative to the water. Find (a) the crossing time, the drift and the boat's velocity relative to the bank if it is pointed straight across, and (b) the heading required to land directly opposite the start, and the crossing time for that heading. Take $+x$ downstream and $+y$ across the river.

Set up the subscripts. $\vec{v}_{BW}$ is the boat relative to the water, $\vec{v}_{WG} = 1.8\hat{i}$ m/s is the water relative to the ground, and $\vec{v}_{BG} = \vec{v}_{BW} + \vec{v}_{WG}$ is what an observer on the bank sees.
(a) Pointed straight across, $\vec{v}_{BW} = 3.0\hat{j}$ m/s, so $\vec{v}_{BG} = 1.8\hat{i} + 3.0\hat{j}$ m/s.
Crossing time uses the $y$ component only, because the current has no $y$ component. $t = 90/3.0 = 30$ s.
Drift is the $x$ component carried for that time. $\Delta x = (1.8)(30) = 54$ m downstream.
Velocity relative to the bank. $\left| \vec{v}_{BG} \right| = \sqrt{(1.8)^2 + (3.0)^2} = \sqrt{12.24} = 3.50$ m/s, at $\tan^{-1}(1.8/3.0) = 31.0^\circ$ from straight across, tilted downstream.
(b) To land directly opposite, the resultant must have no $x$ component, so the boat's own velocity must supply $-1.8$ m/s along $x$ . With the heading $\theta$ measured upstream from straight across, $3.0\sin\theta = 1.8$ , giving $\sin\theta = 0.60$ and $\theta = 36.9^\circ$ upstream of straight across.
What is left for crossing. $v_y = 3.0\cos 36.9^\circ = 3.0(0.800) = 2.40$ m/s, so $t = 90/2.40 = 37.5$ s.
Compare the two. Pointing straight across is the faster crossing, 30 s against 37.5 s, because all 3.0 m/s goes into the crossing. Landing opposite costs 7.5 s, and would be impossible if the current exceeded 3.0 m/s.

(a) 30 s, drifting 54 m downstream, moving at 3.50 m/s at $31.0^\circ$ downstream of straight across. (b) Head $36.9^\circ$ upstream of straight across, crossing in 37.5 s. AP Physics 1's Topic 1.4 boundary statement puts this whole question out of its scope.

Two cars on perpendicular roads

Car A drives east at 20 m/s and car B drives north at 15 m/s, both at constant velocity, measured from the ground. Find the velocity of A relative to B, and the velocity of B relative to A. Then state what each driver measures for the other's acceleration.

Take $+x$ east and $+y$ north. In the ground frame, $\vec{v}_{AG} = 20\hat{i}$ m/s and $\vec{v}_{BG} = 15\hat{j}$ m/s.
Use the chain with a reversal. $\vec{v}_{AB} = \vec{v}_{AG} + \vec{v}_{GB} = \vec{v}_{AG} - \vec{v}_{BG} = 20\hat{i} - 15\hat{j}$ m/s.
Magnitude and direction. $\left| \vec{v}_{AB} \right| = \sqrt{(20)^2 + (15)^2} = \sqrt{625} = 25$ m/s, at $\tan^{-1}(15/20) = 36.9^\circ$ south of east.
The reverse. $\vec{v}_{BA} = -\vec{v}_{AB} = -20\hat{i} + 15\hat{j}$ m/s, the same 25 m/s at $36.9^\circ$ north of west.
Check against the trap. The speeds are 20 and 15 m/s and the relative speed is 25 m/s, larger than either. Adding or subtracting the speeds would have given 35 or 5 m/s, and both are wrong because the motions are perpendicular.
Accelerations. Both cars move at constant velocity, so every acceleration in the ground frame is zero, and by 1.4.B.2.ii each driver measures zero for the other. Each sees the other moving in a straight line at a steady 25 m/s.

$\vec{v}_{AB} = 20\hat{i} - 15\hat{j}$ m/s, which is 25 m/s at $36.9^\circ$ south of east; $\vec{v}_{BA}$ is the same speed at $36.9^\circ$ north of west. Both drivers measure zero acceleration for the other.

A ball dropped in a moving train, from two frames

A train moves horizontally at a constant 12 m/s. A passenger releases a ball from rest, relative to the train, at a height of 1.8 m above the carriage floor. Use $g = 9.8 \ \mathrm{m/s^2}$ . Find the fall time, where the ball lands as measured in each frame, and the ball's acceleration and final velocity in each frame.

Fall time. The vertical motion is identical in both frames because the train's velocity is horizontal, so $1.8 = \frac{1}{2}(9.8)t^2$ gives $t^2 = 0.3673$ s $^2$ and $t = 0.606$ s.
In the train's frame the ball starts at rest, so it has no horizontal velocity and lands at the passenger's feet, straight down from the release point.
In the ground frame the ball is released already moving at 12 m/s horizontally, and nothing acts horizontally, so it keeps that component. Horizontal travel $= (12)(0.606) = 7.27$ m, and the path is a parabola.
Both descriptions are of the same landing point. The carriage floor also moved 7.27 m in that time, so the ball still lands at the passenger's feet. Statement 1.4.A.1 is doing the work here: the choice of frame determines the direction and magnitude of what is measured, and neither observer is wrong.
Final vertical velocity, the same in both frames. $v_y = (9.8)(0.606) = 5.94$ m/s downward.
Final velocity in the train's frame: 5.94 m/s straight down. In the ground frame: $\sqrt{(12)^2 + (5.94)^2} = \sqrt{179.3} = 13.4$ m/s, angled below the horizontal.
Acceleration in both frames. $9.8 \ \mathrm{m/s^2}$ downward, identical, exactly as 1.4.B.2.ii requires, because the train's velocity is constant and so contributes nothing on the second differentiation.

Fall time 0.606 s. The ball lands at the passenger's feet in both descriptions, having travelled 7.27 m horizontally in the ground frame and none in the train frame. Final speed 5.94 m/s in the train frame and 13.4 m/s in the ground frame, but the acceleration is $9.8 \ \mathrm{m/s^2}$ downward in both.

Frequently asked questions

What is the difference between AP Physics C Topic 1.4 and AP Physics 1 Topic 1.4?

One sentence. The titles, both learning objectives and all five essential-knowledge statements are word for word identical, and neither course prints a single equation for this topic. AP Physics 1's boundary statement has a second paragraph that AP Physics C does not print: adding or subtracting vectors to find relative velocities is restricted to motion along one dimension for AP Physics 1. Without that restriction, AP Physics C can ask two-dimensional relative-velocity questions such as a boat crossing a river or an aircraft in a crosswind.

Does AP Physics C test two-dimensional relative velocity, like a boat crossing a river?

Yes. AP Physics 1's Topic 1.4 boundary statement explicitly restricts relative-velocity vector addition to one dimension, and AP Physics C: Mechanics does not print that restriction. Essential knowledge 1.4.B.2.i says combining the motion of an object and the motion of an observer involves the addition or subtraction of vectors, with no limit on the dimensionality, and the Topic 1.5 boundary statement establishes that AP Physics C: Mechanics expects students to quantitatively analyze motion in two dimensions.

How do you find the velocity of one object relative to another?

Subtract the velocities as vectors, in the frame you already have them in. If both are measured from the ground, the velocity of A relative to B is the velocity of A minus the velocity of B, component by component. A subscript convention prevents most errors: write each velocity as of-what-relative-to-what, so that the velocity of A relative to C equals the velocity of A relative to B plus the velocity of B relative to C, with the inner subscripts cancelling. Reversing the order of the subscripts negates the vector.

Why is acceleration the same in all inertial reference frames?

Because the frames differ by a constant velocity, which disappears on the second derivative. If one frame moves at a constant velocity V relative to another, the two velocity measurements of the same object differ by exactly V, and differentiating that difference with respect to time gives zero because V does not change. So both observers measure the same acceleration. This is essential knowledge 1.4.B.2.ii, and the derivation shows why it holds only for inertial frames: if the frame accelerates, the derivative of V is not zero and the two observers disagree.

Should a boat point straight across a river or angle upstream?

It depends which question is being asked. Pointing straight across gives the shortest crossing time, because the boat's whole speed goes into crossing and the current cannot affect a perpendicular component, but the boat lands downstream of its starting point. Angling upstream by enough to cancel the current lands the boat directly opposite, but leaves less of its speed for crossing, so it takes longer. If the current is faster than the boat's speed relative to the water, landing directly opposite is impossible.

What does it mean to assume a reference frame is inertial?

It means assuming the frame is not accelerating, so that acceleration measurements made in it agree with those from any other inertial frame. The AP Physics C Topic 1.4 boundary statement says that unless otherwise stated, the frame of reference of any problem may be assumed to be inertial, and the exam-conventions box on the Table of Information repeats it as the first of its three conventions. In practice you only need to think about it when a question deliberately puts an observer inside something that is accelerating, such as a braking car or a lift starting to move.

Are there any equations for AP Physics C Topic 1.4?

The CED prints none for this topic in either AP Physics C: Mechanics or AP Physics 1, and nothing on the AP Physics C Table of Information corresponds to it. Relative motion is stated in sentences: the observed velocity of an object results from the combination of the object's velocity and the velocity of the observer's reference frame, and combining them involves the addition or subtraction of vectors. The working relationship most people write down, that the velocity of A relative to C is the velocity of A relative to B plus the velocity of B relative to C, is a convention for organising that vector sum rather than a printed formula.