6. (3 points) In this question we investigate how to model a "gravity assist". This is a clever maneuver performed by spacecraft traveling across the solar system in order to save sub- stantial amounts of fuel. The idea is to aim the spacecraft towards a planet or large asteroid so that the spacecraft narrowly misses crashing into the planet. The spacecraft will then be deflected by the gravitational force of the planet, as illustrated in Figure 5. This can be used to change the velocity of the spacecraft (in magnitude or direction) without needing to use up fuel. For example, the Cassini-Huygens spacecraft, which has been sent to Saturn and its moons, traveled for almost 7 years across the solar system, receiving two gravity assists from Venus, one from the Earth, and one from Jupiter. Then, while in orbit in the Saturn system, Cassini performed several gravity assists using Saturn's moon, Titan, allowing it to explore the system from a large variety of orbits and vantage points. Figure 5: Example of a gravity assist. A spacecraft of mass, m, and initial velocity, v, is deflected by a planet of mass, M, and velocity, u. The velocities are as measured in the frame of reference of the Sun. Page 2 (a) Explain why a gravity assist, as in Figure 5, can be modeled as an elastic collision between the spacecraft and the planet. Assume that there are no other bodies exerting any forces of gravity on the two objects, and that the initial and final velocities (before and after the collision) are evaluated when the spacecraft and planet are far apart from each other, when they no longer exert any significant force of gravity on each other.(b) Consider a gravity assist to deect the velocity of the spacecraft by 90 using a planet of mass, Zl/I, that is initially stationary (a : 0), as depicted in Figure 6. Model this as an elastic collision where the initial and nal velocities correspond to their values when the spacecraft and planet are far apart to: 0 determine the speed of the spacecraft, c', after the collision in terms of the speed of the spacecraft before the collision, In, and the masses of the spacecraft, m, and of the planet, 1M. 0 show that in the limit when the mass of the planet, M' , is much bigger than that of the spacecraft, m, (so that a 0), the speed of the spacecraft remains unchanged (1; : o'). F gure 6: Example of a gravity assist to deflect the spacecraft by 90. (c) Consider a gravity assist that is similar to a headon collision between the planet and spacecraft, as illustrated in Figure 7. In this case the planet has an initial velocity, 11', before the collision. Again, model this as an elastic collision where the initial and nal velocities correspond to their values when the spacecraft and planet are far apart to: 0 determine the speed of the spacecraft, v' , after the collision in terms of the speed of the spacecraft, v, and of the planet, a, before the collision, and of the masses of the spacecraft, m, and of the planet, M'. 0 show that in the limit when the mass of the planet, M', is much bigger than that of the spacecraft, m, (so that g > 0), the speed of the planet after the collision is unchanged (u' : u), and that the speed of the spacecraft is \"U, : v + 2a (where u is a negative number if using the diagram in Figure 7). Figure '7: Example of a gravity assist to deflect the spacecraft by 180. (d) Consider the gravity assist depicted in part c) (Figure 7), when the mass of the planet, MT, is very large compared to that of the probe, so that the Speed of the probe after the collision has magnitude: M = lvl + 2lul The Cassini probe has a mass of m : 5000 kg, and the orbital speed of Jupiter is approximately a : 13 km/ s. If the Cassini probe approaches Jupiter with a Speed 12 = 30 km/s (the orbital speed of Earth), how much kinetic energy does it gain from the gravity assist? Give your answer in Joules and determine how many kilograms of hydrogen (rocket fuel) would have to be used to provide the same amount of energy (hydrogen can provide about 140 x 106 J per kilogram)