Exercise:
An observer on Earth sees an alien vessel approaching at a speed of .c. The Hunter spaceship comes to the rescue overtaking the aliens while moving directly toward Earth at a speed of .c relative to Earth. What is the relative speed of one vessel as seen by the other?

Solution:
center tikzpicturescale. scope%Earth filldrawcolorblue fillblue!!white circle cm; drawblulatex --; drawblulatex --; nodeblue at .. mathcalS'; drawcolorred thick stealth - -- nodeabove midway v.c; scope scopescale. xscal xshift-cm yshiftcm %flaps coordinate P at --; drawfillblue!!white - tooutin -. tooutin -. tooutin - -- -; scoperotate around:Pshift-cm-cmyscalexscal drawfillblue!!white - tooutin -. tooutin -. tooutin - -- -; scope %rocket drawfillblack!!white -- tooutin - tooutin tooutin --; %peak drawfillblue!!white -. tooutin . tooutin tooutin -.; %central flap drawfillblue!!white - tooutin - - tooutin -; %porthole drawcolorgrayultra thickfillwhite!!blue -. circle .cm; %fuel foreach i in ... pgfmathsetmacroxrand*. + *- - . pgfmathsetmacroyrand*. + *.-. pgfmathsetmacroopacValrand*.+ drawcolorred fillred!!yellow xy circle .cm; drawred-latex ---; drawred-latex --; nodered at - mathcalS; drawcolorblue thick stealth - -- nodeleft -v-.c; scope scopescale. xscal xshift-cm yshift-cm %flaps coordinate P at --; drawfillblue!!white - tooutin -. tooutin -. tooutin - -- -; scoperotate around:Pshift-cm-cmyscalexscal drawfillblue!!white - tooutin -. tooutin -. tooutin - -- -; scope %rocket drawfillyellow!!white -- tooutin - tooutin tooutin --; %peak drawfillblue!!white -. tooutin . tooutin tooutin -.; %central flap drawfillblue!!white - tooutin - - tooutin -; %porthole drawcolorgrayultra thickfillwhite!!blue -. circle .cm; %fuel foreach i in ... pgfmathsetmacroxrand*. + *- - . pgfmathsetmacroyrand*. + *.-. pgfmathsetmacroopacValrand*.+ drawcolorred fillred!!yellow xy circle .cm; drawcolorblue thickstealth- -- nodeleft u'-.c; drawcolorred thickstealth- ---- noderight u; scope tikzpicture center The relative speed of the two vessels as seen on Earth is of course .c-.c.c. In the reference frame of the alien vessel the Earth is moving at -.c i.e. towards the alien vessel and the Hunter vessel is moving +.c relative to the Earth. Thus the speed of the Hunter as seen from the alien vessel is u fracv+u'+fracvu'c^ frac-.c+.c+-.. .c