1. You discover a previously unknown asteroid in the asteroid belt, and you find that it is in an approximately circular orbit at a distance of 3.3 AU from the Sun. What is the orbital period of this asteroid? Provide your answer in units of years. 2. In class, we discussed the how we can derive orbital speed with Newton's Laws. Newton's Laws tell us that for an object orbiting in a circular orbit around a much more massive object (such as a planet orbiting around the Sun), the orbital speed, v, is given by v = 4/ GTM, where M is the mass of the more massive object, and R is the orbital radius. Given that Earth's orbital speed around the Sun is 30 km/s, determine the orbital speed of the asteroid from Problem 1. Provide your answer in units of km/s. Hint: This is a problem that you can work out by setting up a ratio. 3. Determine an equation for orbital speed, v, that depends on the semi-major axis, a, and the orbital period, P. Recalculate v with this equation (in units of km/s). Does your new result agree with the answer you got using the equation of orbital speed derived from Newton's Laws? Hint: Start with the definition of velocity. 4. Suppose you weigh 150 pounds on Earth. What would your weight be if you were standing on the surface of Pluto? Your answer should be in units of pounds. You can find the physical properties of Pluto in the Appendix. Hint: See Example 3.3 in the textbook for guidance. EXAMPLE 3.3 Calculating Weight By what factor would a person's weight at the surface of Earth change if Earth had its present mass but eight times its present volume? Solution With eight times the volume, Earth's radius would double. This means the gravitational force at the surface would reduce by a factor of (1/2)? = 1/4, so a person would weigh only one-fourth as much. Check Your Learning By what factor would a person's weight at the surface of Earth change if Earth had its present size but only one-third its present mass? 3+ Orbits and Gravity Answer: With one-third its present mass, the gravitational force at the surface would reduce by a factor of 1/3,so a person would weight only one-third as much