This question is about a scientific satellite in circular orbit around Mars.
Define gravitational potential energy of an object at a point in a gravitational field.
The satellite has mass 1200 kg1200\text{ kg}1200 kg. The orbital radius of the satellite around Mars is 2.0×107 m2.0 \times 10^7\text{ m}2.0×107 m. The orbital period of the satellite is 8.60×104 s8.60 \times 10^4\text{ s}8.60×104 s. The mass of Mars is 6.4×1023 kg6.4 \times 10^{23}\text{ kg}6.4×1023 kg. (Take G=6.67×10−11 N m2 kg−2G = 6.67 \times 10^{-11}\text{ N m}^2\text{ kg}^{-2}G=6.67×10−11 N m2 kg−2)
Show that the magnitude of the gravitational potential energy of the satellite is about 2.6×109 J2.6 \times 10^9\text{ J}2.6×109 J.
Show that the kinetic energy of the satellite is half the magnitude of its gravitational potential energy.
Calculate the total energy of the satellite.
The power source for the instrumentation on board the satellite is curium-244, which provides 320 W320\text{ W}320 W initially.
Curium-244 decays by α\alphaα-particle emission with a half-life of 18 years18\text{ years}18 years. The kinetic energy of each α\alphaα-particle is 9.3×10−13 J9.3 \times 10^{-13}\text{ J}9.3×10−13 J. (Take 1 year=3.16×107 s1\text{ year} = 3.16 \times 10^7\text{ s}1 year=3.16×107 s)
Calculate the number NNN of curium-244 nuclei needed to provide the power of 320 W320\text{ W}320 W initially.
Calculate the power PPP still available from the curium-244 source 25 years25\text{ years}25 years later.
