Question

Can someone help me with this question and show all work

1) Planet Velocities and Energy (33 pts) We talked about how planet formation involves the collisions of bodies (planetesimals, embryos) leading to the growth (and heating) of a planet. Let's think about the velocities and energies involved here. a) The speed of a body in its orbit around the Sun is given by the equation: Here Vis the speed of the body in m/s, G is the gravitational constant, Me is the mass of the Sun in kilograms, a is the semi-major axis of the body in meters, and ris its current distance from the Sun in meters (remember, in an elliptical orbit, the distance between the Sun and the body is always changing). Assume you have an asteroid with a semi- major axis of 3 AU and a perihelion of 1 AU (so it's potentially an Earth crossing orbit). What would its speed be at 1 AU? If the Earth's orbital speed is 30 km/s, what is the lowest velocity at which this asteroid could possibly impact the Earth (what is the difference in orbital velocity, if both bodies are orbiting in the same direction)? What if it was a comet, which has a semi-major axis of 20 AU and a perihelion of 1 AU? (Here we're ignoring the acceleration from Earth's gravity to keep things simple.) b) The accretion of planets can involve very large, energetic impacts. The impact that formed the Moon is typically assumed to have involved the collision of a Mars-sized impactor (about 10% the mass of the Earth) with the Earth at a velocity of 10 km/s. If the kinetic energy of a moving body is given by Ekinetic (1/2)mv2 where m is its mass and v its speed, what was the energy of the kinetic energy of the impactor? Give your answer in Joules (J) where 1 J-1 kg m2/s2. If it takes 200 Joules of energy to increase the temperature of 1 kg of Earth-material by 1 degree Kelvin, what would the average temperature change for Earth's rocks be as a result of this collision? If rocks melt at temperatures of about 1500 K, what would you guess about the state of matter on the planet immediately after the impact? c) What would the semi-major axis of the impactor described in (b) have had to have been in order to strike the Earth with a velocity of 10 km/s using the equation in question (a)?

Answer 1