1. 1
   A crewed lander of mass 1.5 × 10⁴ kg rests on the surface of the Moon. The Moon can be treated as a uniform sphere of mass 7.35 × 10²² kg and radius 1.74 × 10⁶ m, with no atmosphere. Before returning to Earth, the lander must rise to join a command module orbiting 100 km above the lunar surface.

   (a)

   What is the escape speed from the surface of the Moon?

   [1 mark]

   • A1.7 km s⁻¹
   • B5.6 × 10³ km s⁻¹
   • C3.4 km s⁻¹
   • D2.4 km s⁻¹

   (b)

   What is the gravitational potential at the surface of the Moon?

   [1 mark]

   • A+2.8 × 10⁶ J kg⁻¹
   • B−1.6 J kg⁻¹
   • C−2.8 × 10⁶ J kg⁻¹
   • D−4.9 × 10¹² J kg⁻¹

   (c)

   Calculate the minimum work done against gravity to raise the lander from the lunar surface to the height of the command module's orbit.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A geostationary communications satellite orbits above the equator with a period of 24.0 hours, so it stays above the same point on Earth. The Moon orbits Earth with a period of 27.3 days at an orbital radius of 3.84 × 10⁸ m. The mass of Earth is 5.97 × 10²⁴ kg and its radius is 6.37 × 10⁶ m. Treat both orbits as circular.

   (a)

   Using Kepler's third law and the data for the Moon, what is the orbital radius of the geostationary satellite?

   [1 mark]

   • A4.2 × 10⁷ m
   • B1.4 × 10⁷ m
   • C2.7 × 10⁶ m
   • D7.3 × 10⁷ m

   (b)

   The gravitational field strength at Earth's surface is 9.81 N kg⁻¹. What is the gravitational field strength at the orbit of the geostationary satellite?

   [1 mark]

   • A0.033 N kg⁻¹
   • B0.22 N kg⁻¹
   • C1.5 N kg⁻¹
   • D9.8 N kg⁻¹

   (c)

   Show that the orbital speed of a satellite does not depend on its mass, and determine the orbital speed of the geostationary satellite.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A satellite of mass 500 kg is in a circular orbit around Earth (mass 5.97 × 10²⁴ kg). At the start of a 90-day tracking period its orbital radius is 6.78 × 10⁶ m. The very thin upper atmosphere exerts a small drag force on it, and at the end of the tracking period its orbit is still almost circular but its radius has fallen to 6.73 × 10⁶ m.

   (a)

   Calculate the change in the gravitational potential energy of the satellite over the tracking period.

   [3 marks]

   (b)

   The satellite's speed increases during the tracking period, even though drag acts against its motion. Explain this, and determine the energy transferred by the drag force.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A probe falls radially towards an airless, spherical planet of radius 3.0 × 10⁶ m. Its instruments record the gravitational potential at four distances r from the planet's centre. At r = 9.0 × 10⁶ m, V = −4.0 × 10⁶ J kg⁻¹. At r = 6.0 × 10⁶ m, V = −6.0 × 10⁶ J kg⁻¹. At r = 4.5 × 10⁶ m, V = −8.0 × 10⁶ J kg⁻¹. At the surface, V = −1.2 × 10⁷ J kg⁻¹.

   (a)

   Use the data to show that V ∝ 1/r, determine the mass of the planet, and estimate the gravitational field strength at r = 7.5 × 10⁶ m from the potential gradient. Comment on the accuracy of your estimate.

   [6 marks]

   (b)

   Describe the equipotential surfaces and gravitational field lines around the planet and how they are related. Use the data to compare the speed with which the probe would hit the surface, if released from rest at r = 9.0 × 10⁶ m, with the escape speed from the surface.

   [6 marks]

   Total for question 4: 12 marks

End of questions