Question

When a missile is shot from one spaceship towards another, it leaves the first at $0.950c$ and approaches the other at $0.750c$. What is the relative velocity of the two ships?

Final Answer

The ships are receding away from each other with a relative velocity of $0.696c$.

### Solution video

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

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Video Transcript

This is College Physics Answers with Shaun Dychko. A missile is shot from one spaceship towards another, the first spaceship reports that this missile is going away from it with a velocity of 0.950

*c*; the second spaceship reports that the missile is approaching it with a velocity of 0.750*c*so in order to figure out whether these velocities should be labeled with*u prime*or*u*, it depends on which spaceship we have assigned the velocity*v*to. So I have decided to just put it on this first spaceship and it doesn't really matter which one you choose— the algebra's the same either way— so we'll take this to have velocity*v*in which case the first ships report for the velocity of the missile is going to be labeled*u prime**u prime*always comes from the reference frame labeled with*v*. So*u prime*is 0.950*c*and that makes*u*0.750*c*. So our formula is that*u*equals*v*plus*u prime*over 1 plus*vu prime*over*c squared*and we want to do some algebra to solve for*v*— the relative velocity of these two spaceships. We'll multiply both sides by 1 plus*vu prime*over*c squared*and then on the left side, after you multiply*u*into this binomial, we'll end up with*u*plus*vu primeu*over*c squared*and on the right hand side, we will have just*v*plus*u prime*. And then subtract*v*from both sides and also subtract*u*from both sides and we end up with*vu primeu*over*c squared*minus*v*equals*u prime*minus*u*. Then factor out the*v*from these two terms and then divide both sides by this binomial*u primeu*over*c squared*minus 1 and you have this formula here that*v*is*u prime*minus*u*over*u primeu*over*c squared*minus 1. So that's 0.950*c*is*u prime*minus*0.750**c*which was*u*divided by the product of those velocities over*c squared*minus 1 and that is negative 0.696*c*. So our answer is negative, which is to say that this first spaceship is moving to the left because we have to the right as positive since*u prime*and*u*are both positive and so our negative means that*v*is to the left and so these spaceships are moving apart and the ships are receding then away from each other with a relative velocity of 0.696*c*.