Predicting Orbits with Mathematical Models — Practice · Grade 9 Science

Chapter: Earth, Moon & Solar System

What this practice covers

By the end of Predicting Orbits with Mathematical Models you should be able to:

8 questions, numeric, multiple choice and written answer, each with a worked explanation.

Vocabulary used: semimajor axis, orbital period, astronomical unit.

Problem 1

For a solar orbit with a = 9 AU, use P² = a³ to find P.
Answer and explanation

27 Earth years

P² = 729; P = 27.

Problem 2

A circular orbit at constant speed has…
  1. zero acceleration
  2. inward acceleration
  3. an outward inertia force balancing gravity
  4. no gravity
Answer and explanation

inward acceleration

Direction changes continuously.

Problem 3

Why cannot P² = a³ in Earth years and AU be applied unchanged to any orbit around any object?
Answer and explanation

The relation’s constant depends on the central mass and chosen units.

Problem 4

For the same star, orbit B has semimajor axis 4 times A's. What is its period ratio? Explain the reasoning in your own words.
Answer and explanation

Orbit B's period is 8 times orbit A's under this model.

T² scales with a³, so the squared period ratio is 4³=64.

Take the square root: T_B/T_A=8.

Response criteria

  • Use the stated conditions.
  • Show the evidence or calculation.
  • Explain the conclusion and any stated limitation.

Problem 5

Using P²=a³, find P when a=9 AU. Show the reasoning, not just the conclusion.
Hints

The relation requires the stated units and central mass.

Answer and explanation

P=√729=27 years.

Problem 6

Why cannot the solar-unit formula be applied unchanged to a moon orbiting a planet?
Hints

State what the supplied observations or conditions establish, and separate that from any assumption.

Answer and explanation

The central mass differs; the numerical relationship and units must be adjusted.

Problem 7

Period grows faster than semimajor axis in the solar relationPeriod grows faster than semimajor axis in the solar relationOrbital period P (years)016324864014916Semimajor axis a (AU)Same dominant solar mass: P² = a³, so P = √(a³).
Original graph of the chapter’s supplied ideal solar-system records. The points follow P² = a³; straight connecting guides do not replace the smooth underlying curve. It is not a dataset of four named planets and is not valid with the same constant for arbitrary central masses.
Interpret the supplied visual model. Find P when a = 9 AU in the supplied solar model.
Answer and explanation

P = √(9³) = √729 = 27 years. The relation assumes the same dominant solar mass and appropriate years/AU units.

Read a on the horizontal axis and P vertically. Neither variable is instantaneous speed or planet diameter.

P = √(9³) = √729 = 27 years. The relation assumes the same dominant solar mass and appropriate years/AU units.

Response criteria

  • Read the labeled quantities, reference frame or objects correctly.
  • Explain the relation between the visual evidence and the mechanism.
  • Keep the drawing’s stated assumptions and limits in the conclusion.

Problem 8

In the ideal same-Sun model, a body has a = 6.25 AU. Find P and the transformed pair (a³, P²). Explain why the same numerical constant need not apply around another star.
Answer and explanation

a³ = 244.140625, so P = 15.625 years and the transformed pair is (244.140625 AU³, 244.140625 years²). The proportional constant depends on the dominant central mass; the convenient solar years/AU form is not universal.

Compute 6.25³ = 244.140625.

Take its square root: 15.625 years.

Keep transformed units and identify the central-mass assumption.

Response criteria

  • Use the stated measurements or conditions, with correct units.
  • Connect the result to the physical mechanism or evidence relationship.
  • Identify the assumption, model limit or further evidence requested.

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