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FRQ: Elliptical Orbits — Comet Aravalli at Perihelion and Aphelion

Assessments aligned to 2026 AP Physics 1 standards

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

▤ Scenario

Comet Aravalli, of mass m, moves around the Sun in a highly elliptical orbit, shown in Figure 1. At its closest point to the Sun, P (perihelion), the comet is a distance r_p = 8.0 × 10¹⁰ m from the Sun’s centre and moves with speed v_p = 5.4 × 10⁴ m/s. At its farthest point, A (aphelion), it is a distance r_a = 1.6 × 10¹² m from the Sun’s centre, twenty times farther. At points P and A, and only at those two points, the comet’s velocity is perpendicular to the line joining the comet to the Sun’s centre.

Figure 1 — Comet Aravalli’s orbit (distances to scale, Sun enlarged), with the points P and A.

The comet interacts only with the Sun. The Sun may be treated as stationary in an inertial reference frame, and all distances are measured from its centre. Treat the comet as a particle of constant mass. Take the comet–Sun system as the system when accounting for gravitational potential energy, and take that energy to be zero when the comet is infinitely far from the Sun.

✎ Free Response Questions

Part A

Indicate whether the comet’s kinetic energy K_a at A is greater than, less than, or equal to its kinetic energy K_p at P by writing one of the following.

•   K_a > K_p

•   K_a < K_p

•   K_a = K_p

Justify your answer using qualitative reasoning beyond referencing equations.

Part B

Starting with conservation of the comet’s angular momentum about the Sun’s centre, derive an expression for the comet’s speed v_a at A. Express your final answer in terms of m, v_p, r_p, r_a, 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

K_a < K_p

The comet interacts only with the Sun, so the only force exerted on it is the Sun’s gravitational pull, a conservative interaction inside the comet–Sun system. The system’s total mechanical energy, the comet’s kinetic energy plus the system’s gravitational potential energy, is therefore the same at every point of the orbit. As the comet travels out from P to A, it moves away from the Sun against the Sun’s attraction, so the system’s gravitational potential energy increases, becoming less negative. The same total then leaves less for kinetic energy, so K_a < K_p. The total mechanical energy stays constant, not the kinetic energy, so what the comet loses in kinetic energy the system gains in gravitational potential energy.

The forces tell the same story (Figure 2). The Sun’s pull on the comet always points toward the Sun. At a point Q partway from P to A, the comet is moving away from the Sun, so part of the pull acts against its motion and slows it. At P and at A the pull is perpendicular to the direction of motion, so at those instants it neither speeds the comet up nor slows it down. The comet slows all the way out from P to A and speeds up all the way back, so it is fastest at P and slowest at A.

Figure 2 — The Sun’s pull on the comet at P, at Q on the way out, and at A.

Scoring (3 points):

▸ A1 — 1 point: For indicating K_a < K_p.

▸ A2 — 1 point: For a justification that indicates one of the following principles: that the total mechanical energy of the comet–Sun system is the same at P and at A; that the comet’s angular momentum about the Sun is the same at P and at A; or that the Sun’s pull changes the comet’s speed only through its component along the comet’s motion (equivalently, through the work it does on the comet).

▸ A3 — 1 point: For a justification that indicates one of the following, for A, where the comet is farther from the Sun: that the gravitational potential energy of the comet–Sun system is greater (less negative) with the comet at A; that at the greater distance the same angular momentum requires a smaller speed; or that as the comet moves away from the Sun between P and A, the pull, directed toward the Sun, acts partly against the comet’s motion (does negative work on it).

Scoring Note: A2 and A3 are content criteria, not route criteria, and they may be earned from different routes. An equation alone, such as E = K + U_G, U_G = −G m_1 m_2 / r or r_p m v_p = r_a m v_a, does not meet A2 or A3; the response must also state in words the physical idea the equation is used to show.

⚠︎ Common error (partial credit): Indicating K_a < K_p because the Sun’s pull on the comet is weaker at A, with no conserved quantity named and nothing said about the pull’s direction relative to the comet’s motion or the work it does — earns 1 of 3 points, A1. The pull at A is indeed 400 times weaker (not 20 times: it falls as 1/r²), but its size at a point does not set the comet’s speed there. At P, where it is 400 times stronger, it is perpendicular to the comet’s motion and is not speeding the comet up at all.

⚠︎ Common error (partial credit): Indicating K_a < K_p because the comet–Sun system loses mechanical energy as the comet moves away from the Sun and slows down — earns 1 of 3 points, A1. The system’s mechanical energy is constant; the kinetic energy the comet loses is stored in the system as gravitational potential energy.

⚠︎ Common error (partial credit): Indicating K_a = K_p because a conserved quantity, the system’s total mechanical energy or the comet’s angular momentum, is the same at P and at A — earns 1 of 3 points, A2. Mechanical energy is conserved as the sum K + U_G, and U_G changes with the comet’s distance from the Sun; angular momentum is conserved as rmv sin θ, which is rmv at P and at A, and r is twenty times larger at A.

⚠︎ Common error (partial credit): Indicating K_a = K_p because the Sun’s pull is perpendicular to the comet’s motion all the way round, so it does no work and cannot change the comet’s speed — earns 1 of 3 points, A2. The pull is perpendicular to the motion only at P and at A; between them it has a component along the motion, backward on the way out and forward on the way back.

⚠︎ Common error (partial credit): Indicating K_a > K_p because the total mechanical energy is constant and the gravitational potential energy, −G m_1 m_2 / r, is smaller (more negative) with the comet farther from the Sun — earns 1 of 3 points, A2. A larger r makes −G m_1 m_2 / r less negative, so U_G is greater at A.

✗ Common error (no credit): Indicating K_a = K_p because a body in orbit moves at constant speed, with nothing said about why — earns 0 of 3 points. That holds only on a circular orbit, where the distance from the Sun never changes; on this orbit the distance changes twentyfold.

Part B — Model Answer

The Sun’s pull on the comet points along the line joining the comet to the Sun’s centre (Figure 2), so it exerts no torque on the comet about the Sun’s centre. With no net torque about that point, the comet’s angular momentum about the Sun’s centre is constant, and its values at P and at A are equal:

L_p = L_a

The magnitude of the comet’s angular momentum about the Sun’s centre is L = rmv sin θ, where θ is the angle between the line from the Sun to the comet and the comet’s velocity (Figure 3). At P and at A, and only there, θ = 90° and sin θ = 1, so

r_p m v_p = r_a m v_a
v_a = v_p r_p / r_a

L = rmv sin θ reduces to rmv only where the velocity is perpendicular to the line from the Sun, at P and at A. The comet’s mass cancels. With the given values, v_a = (5.4 × 10⁴ m/s)(8.0 × 10¹⁰ m)/(1.6 × 10¹² m) = 2.7 × 10³ m/s, one-twentieth of the speed at P.

Figure 3 — The angle θ between the line from the Sun and the velocity: 90° at P and A only.

Scoring (3 points):

▸ B1 — 1 point: For a multistep derivation that includes conservation of angular momentum: an equation setting the comet’s angular momentum about the Sun at P equal to its angular momentum about the Sun at A, in any form. This point is earned for the starting principle alone, even if the steps that follow are incorrect; a final expression for v_a on its own is not a multistep derivation.

▸ B2 — 1 point: For indicating that the comet’s angular momentum about the Sun at P and at A is rmv (L = rmv sin θ with sin θ = 1 there), for example by writing r_p m v_p = r_a m v_a, rmv sin θ at both points with the same angle, which then cancels, or L = Iω with I = mr² and ω = v/r. A reason is not required. A correct final expression for v_a in terms of v_p, r_p and r_a also indicates this relation, unless the response’s own expression for the angular momentum contradicts it.

▸ B3 — 1 point: For a correct expression for the speed at A in terms of v_p, r_p and r_a, v_a = v_p r_p / r_a or an equivalent form, consistent with the indicated expression for the angular momentum. Where B2 was not earned, an expression that follows correctly from the response’s own incorrect expression for the angular momentum also earns this point, provided that expression includes the comet’s distance from the Sun and is used in the same form at P and at A (for example m r² v), and the result is in terms of v_p, r_p and r_a; an incorrect expression with any other error does not.

Scoring Note: A correct, isolated, final expression for v_a earns points B2 and B3. A final expression preceded only by words or by unapplied equations from the reference information is isolated and is not a multistep derivation. Words or equations that state an expression for the angular momentum at P or at A, correct or not, make a final expression not isolated. An expression with the given ratio or values substituted, such as v_p/20, is not a correct expression for v_a, isolated or not; evaluating a correct expression afterwards, as in v_p r_p / r_a = v_p/20 = 2.7 × 10³ m/s, does not lose B3. B1 and B2 carry no consistency credit, and B1, B2 and B3 do not depend on the response in part A.

⚠︎ Common error (partial credit): Writing the comet’s angular momentum as m r² v, the rotational inertia mr² multiplied by the speed instead of by the angular speed v/r, which gives v_a = v_p (r_p / r_a)² — earns 2 of 3 points, B1 and B3. For this comet that is only 1.4 × 10² m/s.

⚠︎ Common error (partial credit): Equating r_p m v_p and r_a m v_a correctly but inverting the ratio when solving, v_a = v_p r_a / r_p — earns 2 of 3 points, B1 and B2. The comet would then move twenty times faster at A than at P.

⚠︎ Common error (partial credit): Writing L_p = L_a but then taking the comet’s angular momentum to be its linear momentum mv, or setting the kinetic energies equal, so that v_a = v_p — earns 1 of 3 points, B1. Angular momentum about the Sun’s centre depends on the comet’s distance from it: at P and at A it is rmv.

✗ Common error (no credit): Starting from conservation of kinetic energy or of linear momentum, with no angular momentum equation, so that v_a = v_p — earns 0 of 3 points. Neither is constant on this orbit. What stays the same is the comet’s angular momentum about the Sun’s centre, on which the pull exerts no torque, and the system’s total mechanical energy.

Part C — Model Answer

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

In v_a = v_p r_p / r_a, the speed at A is the speed at P multiplied by r_p / r_a. At P and A the comet’s speed is inversely proportional to its distance from the Sun, so the farther point is the slower one. Because r_a is greater than r_p, r_p / r_a is less than 1 (it is 1/20), so v_a < v_p, and the comet’s kinetic energy at A, ½ m v_a², is less than its kinetic energy at P, ½ m v_p². That is K_a < K_p, as part A concluded. The two conservation laws agree: part A reached K_a < K_p from the constant mechanical energy of the comet–Sun system, and the equation from part B reaches it from the constant angular momentum of the comet.

(Beyond what part C asks.) The equation also gives the factor: K_a = K_p (r_p / r_a)² = K_p / 400, the same factor by which the Sun’s pull at P exceeds its pull at A, but for a different reason: the kinetic energies compare as (r_p / r_a)² only because the speed at P and at A is inversely proportional to the distance.

Scoring (2 points):

▸ C1 — 1 point: For attempting to address the functional dependence between v_a (or K_a) and the distances r_p and r_a in the equation the response derived in part B.

Scoring Note: It is not necessary to use the functional dependence correctly to earn this point. The response only needs functional-dependence language — such as proportional, inversely proportional, related, increases, decreases, numerator or denominator — to relate v_a (or K_a) and the distances.

▸ C2 — 1 point: For correctly using functional dependence to evaluate how v_a compares with v_p in the equation the response derived in part B, and stating the verdict — consistent or not consistent — that follows from it together with the response’s own part A answer or reasoning. For the correct expression, v_a is smaller than v_p because r_p / r_a is less than 1, so K_a < K_p, which is consistent with a part A answer of K_a < K_p and not consistent with K_a = K_p or K_a > K_p.

Scoring Note: The dependence may be read from the structure of the expression, for example that r_a is in the denominator, or shown by evaluating the expression and stating how the speed changes with the distance. The kinetic energies need not be written out.

⚠︎ Common error (partial credit): Stating how v_a depends on the distances, but giving no verdict on whether this agrees with part A — earns 1 of 2 points, C1.

⚠︎ Common error (partial credit): Reading the dependence backwards in an expression in which v_a decreases as r_a increases (such as v_a = v_p r_p / r_a), so that a larger r_a gives a larger v_a, and calling the equation not consistent with K_a < K_p — earns 1 of 2 points, C1.

✗ Common error (no credit): Stating that the two parts agree because both show the comet moving more slowly at A, or quoting v_a = 2.7 × 10³ m/s against v_p, with no words relating v_a to the distances — earns 0 of 2 points. A consistency argument must say how the equation makes the speed depend on the distance.