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Space Physics: Question 8

Syllabus 6.1.1

Structured Extended 7 marks

Engineers tracking the ice-giant exoplanet Coruna monitor two objects orbiting it: its moon, Tavel, and an uncrewed probe, Ranger-9.

(a) Tavel orbits Coruna at an average orbital radius of 4.50×108 m4.50\times10^{8}\text{ m}, completing one full orbit every 6.00×105 s6.00\times10^{5}\text{ s}. Calculate the average orbital speed of Tavel. [3]

(b) Ranger-9 is placed in a circular orbit around Coruna with an average orbital speed of 3.20×103 m/s3.20\times10^{3}\text{ m/s}, taking 9.00×105 s9.00\times10^{5}\text{ s} to complete one full orbit. Calculate the average orbital radius of Ranger-9's orbit. [4]

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Worked solution

Part (a): Average orbital speed of Tavel

v=2πrTv = \frac{2\pi r}{T}

Distance travelled in one orbit (the circumference): 2πr=2π×4.50×108 m=2.827×109 m2\pi r = 2\pi \times 4.50\times10^{8}\text{ m} = 2.827\times10^{9}\text{ m}

Dividing by the orbital period: v=2.827×109 m6.00×105 sv = \frac{2.827\times10^{9}\text{ m}}{6.00\times10^{5}\text{ s}}

v=4.71×103 m/s (3 s.f.)v = \boxed{4.71\times10^{3}\text{ m/s}}\ \text{(3 s.f.)}

Part (b): Average orbital radius of Ranger-9

Rearranging v=2πrTv = \dfrac{2\pi r}{T} for rr:

r=vT2πr = \frac{vT}{2\pi}

Substituting the values for Ranger-9:

r=3.20×103 m/s×9.00×105 s2πr = \frac{3.20\times10^{3}\text{ m/s} \times 9.00\times10^{5}\text{ s}}{2\pi}

r=2.880×109 m6.283r = \frac{2.880\times10^{9}\text{ m}}{6.283}

r=4.58×108 m (3 s.f.)r = \boxed{4.58\times10^{8}\text{ m}}\ \text{(3 s.f.)}

Final answers

  • (a) Tavel’s average orbital speed == 4.71×103 m/s4.71\times10^3\text{ m/s} (3 s.f.)
  • (b) Ranger-9’s average orbital radius == 4.58×108 m4.58\times10^8\text{ m} (3 s.f.)