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I'm trying to get a better understanding of special relativity and had a few questions I was hoping some helpful forum users could answer. First off, suppose I am moving at 0.5c relative to the earth. If I shine a light directly ahead of me, one would think it would appear to move at 0.5c relative to me because that would make its total velocity relative to earth equal to c. However, according to what I have learned, that does not seem to be true. To me, the light will still appear to be moving at c, which would make its velocity relative to the earth 1.5c, which is impossible. How does the speed of light relate to these difference frames of reference? Secondly, it is my understanding that mass (or apparent mass) increases as matter gets closer to the speed of light. There seems to be a contradiction when this is in a different reference frame, however. Say a spaceship is traveling at .99c relative to earth. Now say a smaller ship comes out of its cargo bay and starts to accelerate. In its frame of reference, the smaller ship will appear to be accelerated from a standstill, so should not have to worry about increasing mass until it gets closer to its own relative speed of light. However, from earth's frame of reference, it is already moving at extreme relativistic speeds so should find it nearly impossible to accelerate and encounter an asymptote with diminishing returns as it approaches the speed of light. Which of these situations would actually happen, or is it some weird mix of the two? My brain has been wired for Newtonian mechanics (too much KSP, yada yada) and I am just trying to build a better intuition and grasp on the concepts of special relativity and how it relates to light. Thank you for your help, and please don't be (too) condescending about my lack of knowledge on the subject!

This is my first post as I have a serious but "heavy" question regarding the recent news from NASA about TRAPPIST-1's planets. I posted on Facebook, to my friends, the following: "This particular system is 40 light years away, meaning even if we could travel at light speed it would take 40 years to reach it. However, realize that as you began your journey and say, for example, you can see it from Earth or your space ship telescope--you're looking at a planet 40 years in the past. As you travel toward the planet and periodically look at it again, do you know how how much time would have passed on that planet, say, ten years into your journey? Well, ten years would pass, but are you SEEING it ten years older? How much older would the planet be by the time you arrive?" Wondering on my own questions, I want to know if I'm on the right track. The planet will not have an 40 nor an 80 years difference of age, but somewhere in between, as the distance of space between yourself and the planet would be getting smaller and smaller. But does anyone have an idea of how I could figure out the exact age of the planet by the time our hypothetical traveler arrives? What equation(s) would be used in this physics question?