How to find the velocity of two objects moving apart at relativistic speeds?

Question

2 identical spacecraft leave Earth at the same velocity but in different directions. If the velocity of each spacecraft was 0.84c, what value would an observer on Earth obtain when he calculated their relative velocities?

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Answer 1 · 2018-08-04T12:24:15

the earth observer sees both spacecrafts moving away at #0.84c#
the one spaceship sees the other at #~=##0.985c#

γ=#1/(sqrt(1-(u²)/(c²)))# is the lorenz transformation

#x'=(x-ut)γ# (1)
#y'=y#
#z'=z#

#t'=(##t-##(ux)/(c²)#)#γ# (2)

rewrite 1 and 2 as

#dx'=(dx-udt)γ#

#dt'=(##dt-##(udx)/(c²)#)#γ#

now divide them

#(dx')/(dt')#=#u_x'#=#v'#= #[(dx-udt)c²]/(c²dt-udx)#

multiply by #1/(dt)# up and down

#v'#=#[(v-u)c²]/(c²-uv)#

#v'#=#[v-u]/[1-(uv)/(c²)]#

now since they move in opposite directions set
v=0.84c and u=-0.84c

and you get
#v'#=#[1.68c]/[1+(0.7056c²)/(c²)]## ##~=##0.985c#

I hope this helped!