13.60 ... Mass of a Comet. On July 4, 2005, the NASA space- craft Deep Impact fired a projectile onto the surface of Comet Tempel 1. This comet is about 9.0 km across. Observations of sur- face debris released by the impact showed that dust with a speed as low as 1.0 m/s was able to escape the comet. (a) Assuming a spherical shape, what is the mass of this comet? (Hint: See Example 13.5 in Section 13.3.) (b) How far from the comet's center will this debris be when it has lost (1) 90.0% of its initial kinetic energy at the surface and (ii) all of its kinetic energy at the surface?

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Steve
To determine:(a) The mass of the comet.(b) (i) The distance between comet’s center and debris when debris has lost of initial kinetic energy at the surface.(b) (ii) The distance between comet’s center and debris when debris has lost all of its initial kinetic energy at the surface.Answer:(a) The mass of the comet is.(b) (i) The distance between comet’s center and debris when debris has lost of initial kinetic energy at the surface is.(b) (ii) The distance between comet’s center and debris when debris has lost all of its initial kinetic energy at the surface is infinity.Explanation:(a)The expression for the mass of the comet is as follows,Here, is the escape velocity of the comet, is the radius of the comet, is the mass of the comet and is the universal gravitational constant.Substitute for , for and for.Therefore, the mass of the comet is.(b)(i)The expression for the initial kinetic energy of the debris is as follows,Here, is the initial velocity of the debris, is the mass of the debris and is the initial kinetic energy of the debris.The expression for the initial gravitational potential energy of the debris is as follows,Here, is the initial gravitational potential energy of the debris.The expression for the final gravitational potential energy of the debris is as follows,Here, is the final gravitational potential energy of the debris and is the distance between comet’s center and debris.The expression for the conservation of energy is as follows,Here, is the final kinetic energy.Substitute for, for , for and for.Substitute for , for , for and for.Therefore, the distance between comet’s center and debris when debris has lost of initial kinetic energy at the surface is.(b)(ii)The debris does not lose all of its initial kinetic energy. But as the distance between comet’s center and debris increases, the kinetic energy of the debris decreases.Therefore, the distance between comet’s center and debris when debris has lost all of its initial kinetic energy at the surface is infinity.

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