Prove that for any relative velocity vv between two observers, a beam of light sent from one to the other will approach at speed cc (provided that vv is less than cc, of course).
Question by OpenStax is licensed under CC BY 4.0
Final Answer

Please see the solution video.

Solution video

OpenStax College Physics, Chapter 28, Problem 32 (Problems & Exercises)

OpenStax College Physics, Chapter 28, Problem 32 (PE) video thumbnail

In order to watch this solution you need to have a subscription.

Start free trial Log in
vote with a rating of votes with an average rating of .
Video Transcript
This is College Physics Answers with Shaun Dychko. We are going to show that when two observers are approaching each other with a relative velocity of v that when light is emitted from one observer with a speed of c of course, we are going to show using this formula that the other observer will also report a velocity for this light as c. So suppose this first observer is labeled with the velocity v that means the velocity for the light that they report is labeled u prime and we are told u prime is c so the question is what velocity does the second observer report for this light? So the velocity that the second observer reports is going to be v plus u prime over 1 plus v times u prime over c squared. So u prime is c so we plug in c, wherever we see u prime here and this vc over c squared becomes just v over c and then we can multiply top and bottom by c and on the top, let's just leave it outside of some brackets and on the bottom, the c gets distributed into this binomial and so we have c plus c times v over c, which is just v and this numerator and denominator are the same so this fraction is 1 and this ends up being c and so we have shown that this second observer will report the same thing as the first observer when they are talking about the speed of light.