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FRQ: Work Done by Two Forces at Constant Speed — The Slow Spring Pull

Assessments aligned to 2026 AP Physics 1 standards

Qualitative/Quantitative Translation (QQT)  |  CHALLENGE  |  8 points  |  20 min

▤ Scenario

A block of mass m rests on a frictionless horizontal surface and is attached to one end of an ideal spring of spring constant k; the other end of the spring is fixed to a wall, as shown in Figure 1. The position x of the block is measured from the position at which the spring is relaxed, with the positive direction pointing away from the wall.

A student pulls the block away from the wall with a horizontal cord. The block passes through x = 0 moving with speed v_0, and the student adjusts the pull so that the block's speed remains equal to v_0 as the block moves from x = 0 to x = d. Let W_H be the work done on the block by the cord and W_S be the work done on the block by the spring, both for the motion from x = 0 to x = d.

All motion is one-dimensional. Use the surface as the reference frame.

Figure 1: The block at x = d, moving at constant speed v_0 while the cord pulls it away from the wall.

✎ Free Response Questions

Part A

Indicate whether W_H is greater than, less than, or equal to the magnitude |W_S| of the work done by the spring by writing one of the following.

• W_H > |W_S|

• W_H < |W_S|

• W_H = |W_S|

Justify your answer using qualitative reasoning beyond referencing equations.

Part B

Starting with the work–energy theorem, derive an expression for the work W_H done on the block by the cord as the block moves from x = 0 to x = d. Express your final answer in terms of k, d, 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

W_H = |W_S|

The block's speed does not change from x = 0 to x = d, so its kinetic energy does not change: the total energy transferred to the block by all forces is zero. The normal and gravitational forces act perpendicular to the motion and transfer no energy, so whatever energy the cord puts into the block, the spring must take out. Because the velocity is constant, the cord's pull and the spring's pull are equal in magnitude and opposite in direction at every position, so over each small step the two forces do equal and opposite work. A force doing positive work on the block does not mean the block gains kinetic energy; only the net work does that, and here it is zero.

Scoring (3 points):

▸ A1 — 1 point: Selects W_H = |W_S|.

▸ A2 — 1 point: States that the block's kinetic energy (or speed) does not change, so the total energy transferred to the block by all forces is zero — the cord's work and the spring's work must cancel.

▸ A3 — 1 point: States that because the block moves at constant velocity the cord's pull balances the spring force at every position (equal magnitude, opposite direction), so over each step the cord transfers in what the spring transfers out; or that the pull must increase as the stretch increases.

⚠︎ Common error (partial credit): Selects W_H = |W_S| and says only that "the two works cancel" without linking this to the unchanging kinetic energy or to the balanced forces — earns A1 only, 1 of 3 points.

✗ Common error (no credit): Selects W_H > |W_S| because "the cord does positive work, so the block gains energy": the block's kinetic energy is constant — the sign of one force's work does not fix the sign of ΔK.

Part B — Model Answer

Begin with the work–energy theorem, taking the block as the object:

ΔK = ΣW_i = W_H + W_S

The normal and gravitational forces do no work because they are perpendicular to the displacement. The speed is constant, so ΔK = 0 and W_H = −W_S. The spring force F_s = −kx changes with position, so W_S is the area under the graph of F_s against x from x = 0 to x = d: a triangle of base d and height kd that lies below the axis, because the force points toward the wall while the block moves away from it:

W_S = −½ (d)(kd) = −½kd²

W_H = −W_S = +½kd²

Figure 2: From x = 0 to x = d the cord's work (above the axis) and the spring's work (below) are equal and opposite.

The cord's pull is not constant — it grows from zero to kd as the spring stretches — so its work is the area under F_H = kx, which is ½(kd)(d), not kd × d. Neither m nor v_0 appears in the answer: the kinetic-energy terms cancel exactly.

Scoring (3 points):

▸ B1 — 1 point: Begins with the work–energy theorem, ΔK = ΣW_i (or W_net = ΔK), and sets ΔK = 0 for the constant-speed motion.

▸ B2 — 1 point: Obtains the spring's work as the area under the F_s–x graph, W_S = −½kd² (or, equivalently, the cord's work as the area under F_H = kx).

▸ B3 — 1 point: Arrives at W_H = +½kd².

Scoring Note: A correct, isolated, final expression for W_H earns points B2 and B3.

⚠︎ Common error (partial credit): Treats the pull as a constant force equal to its final value, W_H = (kd)(d) = kd² — earns B1 only, 1 of 3 points.

⚠︎ Common error (partial credit): Keeps a kinetic-energy term that should have cancelled, such as W_H = ½kd² + ½mv_0² — earns B1 and B2, 2 of 3 points.

✗ Common error (no credit): Writes W_H = −½kd²: the cord pulls in the direction the block moves, so its work is positive; the negative area belongs to the spring — earns B1 only.

Part C — Model Answer

W_H = +½kd² is positive and equals the magnitude of the spring's work, ½kd², so the equation gives W_H = |W_S|, as selected in part A. The derivation shows why: with ΔK = 0 the work–energy theorem forces the two works to be equal and opposite — part A's "energy in equals energy out". On the F–x graph the two triangles are mirror images — equal area, opposite sign.

Scoring (2 points):

▸ C1 — 1 point: Compares the magnitude of the expression derived in part B with the comparison selected in part A and correctly states whether the two agree.

▸ C2 — 1 point: Explains the agreement physically — ΔK = 0 requires W_H and W_S to be equal and opposite, or balanced forces give mirror-image areas.

Scoring Note: Point C1 may be earned by consistency with incorrect answers in parts A and B — a student who selected W_H > |W_S| and derived W_H = kd² earns C1 for noting that those results agree. C2 requires correct physics.

✗ Common error (no credit): Claims consistency (or inconsistency) without comparing part B's magnitude with part A's selection — earns nothing.

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