1. 1
   Io and Europa are two of the large moons of Jupiter. Both move in almost circular orbits. Io has an orbital radius of 4.22 × 10⁸ m and an orbital period of 1.77 days. Europa has an orbital radius of 6.71 × 10⁸ m. The mass of Io is 8.93 × 10²² kg and the mass of Europa is 4.80 × 10²² kg. Both moons can be treated as point masses.

   (a)

   What is the orbital period of Europa?

   [1 mark]

   • A1.11 days
   • B2.81 days
   • C3.55 days
   • D4.47 days

   (b)

   What is the ratio (gravitational force of Jupiter on Europa) / (gravitational force of Jupiter on Io)?

   [1 mark]

   • A0.34
   • B0.21
   • C0.40
   • D0.54

   (c)

   Determine the mass of Jupiter.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Halley's comet moves around the Sun in a highly elliptical orbit. At its closest approach (perihelion) it is 0.59 AU from the Sun; at its furthest point (aphelion) it is 35 AU from the Sun. It takes 76 years to complete one orbit. Only the Sun's gravitational force acts on the comet.

   (a)

   According to Kepler's laws, where in its orbit does the comet move fastest?

   [1 mark]

   • AAt aphelion, where it is furthest from the Sun
   • BAt the two points midway between perihelion and aphelion
   • CNowhere; its speed is constant around the orbit
   • DAt perihelion, where it is closest to the Sun

   (b)

   What is the ratio (gravitational field strength of the Sun at perihelion) / (gravitational field strength of the Sun at aphelion)?

   [1 mark]

   • A3.5 × 10³
   • B7.7
   • C59
   • D2.1 × 10⁵

   (c)

   Explain, with reference to Kepler's second law, why the comet spends most of its 76-year orbit far from the Sun.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A student uses published data for the gravitational field strength g of Earth at different distances r from Earth's centre. At r = 6.37 × 10⁶ m (Earth's surface), g = 9.81 N kg⁻¹. At r = 7.00 × 10⁶ m, g = 8.13 N kg⁻¹. At r = 1.00 × 10⁷ m, g = 3.98 N kg⁻¹. At r = 4.22 × 10⁷ m, g = 0.224 N kg⁻¹. The International Space Station (ISS) orbits at a height of 420 km above Earth's surface.

   (a)

   Show that the data are consistent with Newton's inverse-square law of gravitation, and determine the mass of Earth.

   [3 marks]

   (b)

   A student claims that astronauts on the ISS float because there is no gravity at that height. Evaluate this claim, using the data.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A space agency plans a mission to Mars, which is very nearly spherical with a mass of 6.42 × 10²³ kg and a radius of 3.39 × 10⁶ m. A probe will first move in a circular orbit 400 km above the surface of Mars. Later it will hover 5 km above the surface of Phobos, a small, irregular, potato-shaped moon of Mars measuring about 27 km × 22 km × 18 km, with a mass of 1.07 × 10¹⁶ kg. Phobos orbits Mars at a distance of 9.38 × 10⁶ m from the centre of Mars.

   (a)

   Show that, for a body in a circular orbit, Newton's law of gravitation leads to Kepler's third law, and calculate the orbital period of the probe in its orbit around Mars.

   [6 marks]

   (b)

   Discuss whether Newton's law, with the bodies treated as point masses, can be used accurately (i) for the force of Mars on Phobos and (ii) for the force of Phobos on the probe when it hovers 5 km above Phobos's surface. Refer to the gravitational field lines of each body in your answer.

   [6 marks]

   Total for question 4: 12 marks

End of questions