1. Planet Kerzaa is 12 ly from Earth. Twins Bella and Carlos are 20 years old. Bella travels to Kerzaa at 0.6c, very quickly turns around, and returns to Earth at 0.8c. How old will Bella and Carlos be when Bella gets back? [Note: Bella changes her velocity relative to the inertial rest frame of Earth/Carlos (S) when she makes a U-turn at Kerzaa. Therefore, her own inertial rest frame changes at that point. Bella's first rest frame (S'), before reaching Kerzaa, is moving at speed 0.6c from Earth to Kerzaa. Her second rest frame (S"), after reaching Kerzaa and U-turn, is moving at speed 0.8c from Kerzaa to Earth.] 2. Arthur and Beatrice are both born just as Arthur's spaceship passes Earth at 0.9c. Ac- cording to Beatrice on Earth, planet Caldi is a fixed 30 ly away and at rest relative to Earth. As Arthur passes Caldi on his continuing outward journey, what will be: (a) Beatrice's age according to Beatrice, (b) Beatrice's age according to Arthur, (c) Arthur's age according to Arthur, and (d) Arthur's age according to Beatrice? 3. The light from galaxy NGC 221 consists of a recognizable spectrum of wavelengths. However, all are shifted toward the shorter-wavelength end of the spectrum. In partic- ular, the calcium "line" ordinarily observed at 396.85 nm (from a source at rest on Earth), is observed at 396.58 nm, in the light coming from NGC 221. (a) Is the galaxy moving toward or away from Earth? (b) At what speed? 4. With reckless disregard for safety and the law, you set your high-performance rocket cycle on course to streak through an intersection at top speed. Approaching the inter- section, you observe green (540 nm) light coming from the traffic signal. After passing through, you look back to observe red (650 nm) light. Actually, the traffic signal never changed its color when you passed it-it didn't have time! (Note: you are supposed to assume here that this traffic light sends out its light waves with the same wavelength in all directions, so that you can observe those light waves both before and after you passed the intersection with your rocket cycle.) (a) What is the top speed of your rocket cycle? (b) What was the color of the traffic signal (according to an appalled bystander)?