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FRQ: Displacement Components in Two Dimensions — The Sonar-Tracked AUV

Assessments aligned to 2026 AP Physics 1 standards

Qualitative/Quantitative Translation (QQT)  |  MID-LEVEL  |  8 points  |  20 min

▤ Scenario

A research team tracks an autonomous underwater vehicle (AUV) using a fixed array of sonar transponders mounted on the walls of a rectangular test tank. The transponders report the vehicle’s position every 2.0 s. Figure 1 gives the AUV’s x- and y-coordinates at four successive times during one run.

Figure 1 — The AUV’s coordinates in the tank frame at four successive transponder readings.

The AUV’s buoyancy system holds it at a constant depth for the whole run, and its propellers maintain a steady thrust. Positive x points east and positive y points north. All coordinates are measured in the tank frame, which may be treated as inertial. The motion is two-dimensional, in a single horizontal plane.

✎ Free Response Questions

Part A

Let θ_1 be the angle between due east and the AUV’s displacement over the first interval, from t = 0.0 s to t = 2.0 s, and θ_3 the angle between due east and its total displacement over all three intervals, from t = 0.0 s to t = 6.0 s.

Indicate whether θ_3 is greater than, less than, or equal to θ_1 by writing one of the following.

•   θ_3 > θ_1

•   θ_3 < θ_1

•   θ_3 = θ_1

Justify your answer using qualitative reasoning beyond referencing equations.

Part B

Suppose an AUV in the same tank makes n successive displacements, each with x-component s_x, where s_x > 0, and y-component s_y, where s_y < 0. Let θ_n be the angle between due east and its total displacement over the n displacements.

Starting with the relationship between the direction of a vector and its perpendicular components, derive an expression for θ_n. Express your final answer in terms of n, s_x, s_y, and physical constants, as appropriate. Begin your derivation by writing the fundamental physics principle or an equation from the reference information.

Part C

Justify how your derived equation in part B is or is not consistent with your reasoning in part A.

❖ Answer Key & Scoring Guide

▸ earns credit  ⚠︎ common error, partial credit  ✗ common error, no credit

Points are independent and there are no deductions. ▸ criteria follow College Board’s published scoring standard; ⚠︎ and ✗ lines are TheScienceCube teaching commentary, not College Board rubric.

Part A — Model Answer

θ_3 = θ_1

Read Figure 1 one interval at a time, subtracting each reading from the next. Between every pair of successive readings the x-coordinate rises by the same 2.0 m and the y-coordinate falls by the same 1.5 m, so the AUV’s displacement in each 2.0 s interval is 2.0 m east and 1.5 m south. It is the same in all three intervals: the same length, pointing the same way.

The total displacement over all three intervals is those three interval displacements placed head to tail, each starting where the one before it ended. Because all three point the same way, they line up along a single straight line through the starting position, and the total displacement runs along that line from the first reading to the last. Its east change of 6.0 m and south change of 4.5 m are each three times those of the first interval, so it is the first interval’s displacement made three times longer, not turned. Steps that all point the same way, laid head to tail, make one longer displacement in that direction, so the angle cannot change.

Figure 2 — Interval displacements head to tail, and the whole-run displacement alongside them.

Scoring (3 points):

▸ A1 — 1 point: For indicating θ_3 = θ_1.

▸ A2 — 1 point: For using the data in Figure 1 to establish that the east and south changes over all three intervals are in the same proportion as those over the first interval.

▸ A3 — 1 point: For a qualitative argument that a displacement whose east and south changes are each the same positive multiple of another’s points in the same direction — for example, that identical interval displacements laid head to tail form one displacement that has not turned.

Scoring Note: A2 and A3 are content criteria, not route criteria. A2 depends only on what the response states about the changes in Figure 1, not on the comparison selected; it may rest on the same changes in every interval, on the whole run’s changes being three times the first interval’s, or on the readings lying in order along one straight line — which also earns A3 where the response concludes that the whole-run displacement lies along the first interval’s line. Evaluating only the two angles, their tangents or their ratios, with no statement about the changes, earns neither A2 nor A3. Stating that changes in the same proportion and sense give the same direction is enough for A3; an argument resting on an incorrect premise, such as a steady thrust fixing the direction of motion, is not.

⚠︎ Common error (partial credit): Indicates θ_3 < θ_1 because in every interval the AUV covers 2.0 m east but only 1.5 m south, so the east total pulls further ahead of the south total — by 0.5 m after one interval and 1.5 m after three — and the eastward motion comes to dominate the southward motion — earns 1 of 3 points, for A2. The angle is set by the proportion between the two totals, which never changes, not by the difference between them. Nor does the eastward motion suppress the southward motion: each component accumulates along its own axis, and the east change is the larger only because each interval’s displacement lies closer to due east than to due south. The AUV’s average velocity in each interval has components +1.0 m/s and −0.75 m/s; the two would be equal in size only for a displacement pointing 45° south of east.

⚠︎ Common error (partial credit): Indicates θ_3 = θ_1 because the propellers maintain a steady thrust, so the AUV must travel in a straight line and its direction never changes — earns 1 of 3 points, for A1. A steady thrust fixes neither the direction of motion nor the path, and the response takes nothing from Figure 1 and says nothing about how the interval displacements combine.

⚠︎ Common error (partial credit): Indicates θ_3 = θ_1 and justifies it only by evaluating tan⁻¹(1.5/2.0) and tan⁻¹(4.5/6.0) and finding both 36.9° — earns 1 of 3 points, for A1. The prompt asks for reasoning beyond referencing equations, and evaluating an equation for each displacement is not that reasoning.

✗ Common error (no credit): Indicates θ_3 > θ_1, arguing that an object moving east and south at the same time follows a curved path, so its direction of travel keeps swinging toward south as the run goes on. Having both an east and a south component does not by itself make the direction change: Figure 1 shows the same 2.0 m east and 1.5 m south change in every interval, so each interval’s displacement points the same way and the total has not swung toward south. The response neither uses the data nor argues how the interval displacements combine, so none of A1 to A3 is met.

Part B — Model Answer

Because θ_n is measured from due east, its tangent is the magnitude of the total displacement’s y-component divided by its x-component:

tan θ_n = |Δy_n| / Δx_n

Each component is a one-dimensional quantity that accumulates along its own axis, so after n displacements

Δx_n = n·s_x
Δy_n = n·s_y

Since s_y is negative, |Δy_n| = −n·s_y. Substituting,

tan θ_n = −n·s_y / (n·s_x) = −s_y / s_x
θ_n = tan⁻¹(−s_y / s_x)

With s_x > 0 and s_y < 0 the ratio −s_y / s_x is positive, so the inverse tangent returns an angle between 0° and 90° — the size of the angle between due east and the displacement, which lies south of east because s_y is negative. The same factor n multiplies both components, so it cancels from their ratio and the angle does not depend on how many displacements are made.

Figure 3 — One displacement and the total of n: both legs scale by n, so θ_n is unchanged.

Scoring (3 points):

▸ B1 — 1 point: For beginning from a correct relationship between the direction of a displacement, measured from due east, and its perpendicular components, applied to the total displacement or to a single displacement — for example tan θ_n = |Δy_n| / Δx_n, or an equivalent sine or cosine relationship. This point is awarded for the correct starting relationship alone, before any algebra is performed.

▸ B2 — 1 point: For expressing the components of the total displacement as n·s_x and n·s_y.

▸ B3 — 1 point: For obtaining a correct expression for θ_n. The expected expression is θ_n = tan⁻¹(−s_y / s_x).

Scoring Note: A correct, isolated, final expression for θ_n earns points B2 and B3. Any expression equal to tan⁻¹(−s_y / s_x) is correct, including tan⁻¹(|s_y| / s_x), sin⁻¹(−s_y / √(s_x² + s_y²)) and cos⁻¹(s_x / √(s_x² + s_y²)); tan⁻¹(s_y / s_x) is also correct, its negative sign marking the angle as south of due east.

⚠︎ Common error (partial credit): Writes tan θ_1 = −s_y / s_x for a single displacement and then θ_n = n·tan⁻¹(−s_y / s_x), adding one displacement’s angle for each displacement, without ever forming the components of the total displacement — earns 1 of 3 points, for B1. The components add, and the angle follows from their ratio; the angles themselves do not add.

⚠︎ Common error (partial credit): Writes the components as n·s_x and n·s_y but takes tan θ_n as the x-component divided by the magnitude of the y-component, obtaining θ_n = tan⁻¹(−s_x / s_y), the angle measured from due south — earns 1 of 3 points, for B2.

✗ Common error (no credit): Reports θ_n ≈ 36.9° and nothing else. The part asks for an expression for θ_n in terms of n, s_x and s_y, and a number evaluated from Figure 1 is not that expression, so no criterion is met.

Part C — Model Answer

The derived equation is consistent with the reasoning in part A.

Figure 1’s three intervals are three such displacements, each with s_x = +2.0 m and s_y = −1.5 m, so θ_1 and θ_3 are θ_n for n = 1 and n = 3. The expression θ_n = tan⁻¹(−s_y / s_x) contains no n, so θ_3 = θ_1, the comparison selected in part A. The components after three intervals are 3(+2.0 m) = +6.0 m and 3(−1.5 m) = −4.5 m, and every θ_n is tan⁻¹(1.5 m / 2.0 m) ≈ 36.9°, south of east.

The equation agrees with the reason part A gave, not only with its answer. The factor n in n·s_x and n·s_y is the part A argument written as algebra: identical steps grow both components together. That lengthens the total displacement without turning it — its magnitude, n√(s_x² + s_y²), is 2.5 m after one interval and 7.5 m after three, while its angle stays at 36.9°.

The part A argument needs only that every interval’s displacement points the same way, with east and south changes in the same proportion and the same sense; the derived equation assumes more, identical components. Had the third interval instead taken the AUV 2.0 m east and 3.0 m south, the proportion would change, the totals would be +6.0 m and −6.0 m, and θ_3 would be 45° rather than 36.9°, so neither would apply. The equation covers only identical steps, one case of the same-direction steps the part A argument covers, so the two agree wherever the equation applies.

Scoring (2 points):

▸ C1 — 1 point: For identifying what the response’s own part B expression implies about how θ_n depends on n, and stating the verdict — consistent, or not consistent — that follows from it together with the response’s own part A selection. For the correct expression θ_n = tan⁻¹(−s_y / s_x) and the selection θ_3 = θ_1, that it contains no n, so θ_3 = θ_1 and the two are consistent.

▸ C2 — 1 point: For connecting the feature of the response’s own part B expression that determines how θ_n depends on n to the response’s own reasoning in part A, showing either that the feature is that reasoning in algebraic form or where that reasoning departs from it. For the correct work, that the absence of n is the algebraic form, for identical displacements, of the part A argument that same-direction displacements add to one longer displacement that has not turned. This point is awarded for work consistent with the response’s own parts A and B, whatever verdict it leads to; explaining why n cancels is not required.

Scoring Note: C1 does not require a statement that the run in Figure 1 fits the part B supposition. C2 may also be earned by identifying that the response’s own part A reasoning holds only while every interval’s displacement points the same way, and that the part B expression, which assumes identical displacements, then fails too — for example by showing that an interval with a different proportion of east to south change would change θ_3.

⚠︎ Common error (partial credit): States that the two are consistent because the derived expression contains no n, but never connects the absence of n to the part A reasoning — earns 1 of 2 points, for C1.

✗ Common error (no credit): Recalculates θ_1 and θ_3 from Figure 1, finds both are 36.9°, and declares the equation consistent without using the part B expression. That checks the part A answer rather than the derived equation, so neither criterion is met.

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