AS 91524 · 2025 paper · momentum & circular motion
Question 1(c) Merit to Excellence
Practises Gravitation & orbital mechanics
Titan is the largest of Saturn's moons. Ignore every other body, and assume Titan is in a perfectly circular orbit.
| M Saturn | 5.68×10²⁶ kg |
|---|---|
| r Saturn | 5.82×10⁷ m |
| M Titan | 1.35×10²³ kg |
| centre-to-centre | 1.22×10⁹ m |
Calculate the time for Titan to complete one orbit of Saturn. You could start with Titan's orbital velocity.
Work it out
Try it on paper first. Stuck? Take one hint at a time.
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1 What force keeps Titan moving in a circle?
Saturn’s gravity, and nothing else. It supplies the whole centripetal force: Fg = Fc. -
2 Write both forces out in full, then solve for Titan’s speed.
GMm / r² = mv² / r. Titan’s mass m cancels, leaving v² = GM / r, so v = √(6.67 × 10⁻¹¹ × 5.68 × 10²⁶ / 1.22 × 10⁹) = 5572.6 m s⁻¹. -
3 How far does Titan travel in one orbit, and how long does that take at this speed?
One lap is the circumference, 2πr. T = 2πr / v = (2π × 1.22 × 10⁹) / 5572.6 = 1.3756 × 10⁶ s. (Strictly, Titan circles the centre of mass from (a), 2.90 × 10⁵ m from Saturn’s centre, but that only changes the fourth figure.)
What the marker wanted
- Achieved One of: gravity set equal to the centripetal force; v = 2πr / T written down; or the gravitational pull between them (about 3.44 × 10²¹ N).
- Merit v² = GM / r reached, by algebra or with the numbers in; or a right method for T from a wrong speed; or T² = 4π²r³ / GM.
- Excellence The right period with its unit, by this route or through v = ωr.
The Achieved column gives the gravitational force as 3.4383 × 10²¹ N, which comes from G = 6.674 × 10⁻¹¹. With the resource booklet’s 6.67 × 10⁻¹¹ it is 3.436 × 10²¹ N. Both are 3.44 × 10²¹ N to three figures.
In our words, from NZQA’s 2025 marking schedule.
How did it go?
Got it moves it further out. Review brings it back tomorrow.