OpenStax College Physics for AP® Courses, Chapter 28, Problem 22 (Problems & Exercises)

This is College Physics Answers with Shaun Dychko. A spacecraft is approaching the Earth with a speed of 0.100c but I didn't label this speed on the spacecraft I put it instead on the Earth because this approach velocity of the two reference frames, which is labeled v can be put on either reference frame so the Earth is a reference frame and the spaceship is a reference frame and they get called reference frames because these are the two places where observers can be to measure the speed of this third thing but there is no observer on the third thing so it doesn't count as a reference frame I mean there could be somebody there but we are not assuming anybody making an observation who is on the message capsule so the observers are on the Earth and the observers are on the spacecraft. So either one of these things— the Earth or the spacecraft— can be labeled with this velocity v and I have chosen to put it on the Earth because we have our formula arranged this way such that the velocity of this message capsule will be the velocity of the reference frame that reports a measurement for the message capsule's velocity and that is the Earth plus what they report on the Earth—u prime— divided by 1 plus v times u prime over c squared. So we are taking right to be the positive direction which makes v negative 0.100c and u prime—the velocity of the message capsule as reported by an Earth-based observer— is negative 0.100c as well. Okay! So really the hard part of these questions is figuring out what object gets labeled with what letter so labeling the message capsule velocity with u prime is important; choosing to put v on the Earth was helpful for making this formula easy to use. Okay! So we have negative 0.100c plus negative 0.100c all divided by 1 plus negative 0.100c times negative 0.100c all over c squared and that is negative 0.198c. And we can ask ourselves whether this is realistic; our false intuition using Galilean relativity would say that the approach velocity of the message capsule according to a person on the spaceship should be negative 0.100 plus negative 0.100 for a total of negative 0.200c and that's wrong but for Galilean relativity that's the way you would do it but at least this relativistic determination should be close to what we expect from our Galilean intuition and it is close to negative 0.200 so there we go!