OpenStax College Physics for AP® Courses, Chapter 28, Problem 26 (Problems & Exercises)

This is College Physics Answers with Shaun Dychko. Two spacecrafts are approaching each other with a speed of 0.800 times c and there's a message canister that is shot out from the first ship towards the second ship and the second ship is measuring the approach velocity of this canister as 0.999c. The hardest part about these questions is to figure out which thing should get labeled with a v and which thing should get labeled with u prime or u. So if we label the second ship with the velocity v and consider the first ship to be at rest then whatever velocity this second ship reports for the message canister, we are going to label that u prime. So u prime is reported by the reference frame that's moving with velocity v; we could have instead labeled the first ship as having velocity v and the second ship being stationary v would then in that case be positive 0.800c because positive is to the right but it would have made our algebra a little different because we would have had to rearrange this to solve for u prime because we don't know what velocity this first ship is going to report— that's the question here— what velocity will the first ship report for this message canister? Okay! So in order to make this first ship reporting velocity u we don't have to do any algebra because now we have it set up so that we have v—this negative 0.800c— because the second ship is going to the left in this picture and u prime is the velocity reported by this second ship, which we are given in the question—0.999c— and so yes, here we go! Plug in some numbers: we have negative 0.800c for v and we have positive 0.999c for the message canister as reported by this second ship, divide by 1 plus negative 0.800c times 0.999c over c squared and this is positive 0.991c. So this first ship will see the message canister received from it with a velocity of positive 0.991c.