# Calculating velocities using Reference frames

Suppose an object A is traveling at a velocity of 100 m/s, and another object B is traveling at 105 m/s. With both the objects traveling through the same direction, taking A as a reference frame, the velocity of B would be 5 m/s (Is this actually right?). But, when they're traveling in opposite directions, how would one measure the velocity of B (with A as reference frame)..? Does it actually take a negative sign? - Sorry, if I have a misunderstanding...

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## 1 Answer

You're quite correct, you'd write the opposite velocity with a negative sign. You just need to decide what sign convention to use. In your example you're only considering motion in one dimension, so you could take motion to the right to be positive in which case motion to the left would be negative. Or you could take motion to the left positive and motion to the right negative. It doesn't matter what convention you use as long as you're consistent.

Suppose we take motion to the right to be positive so in your first example both A and B are moving to the right. The velocity that A measures is $v_B - v_A$, so in your first example it's 105 - 100 or 5 m/s. If B is moving in the other direction the velocity $v_B$ is -105 m/s. It's negative because it's moving to the left. The velocity A measures is now -105 - 100 or -205 m/s. The minus sign tells us the motion is to the left so A sees B moving to the left at 205 m/s.

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Hello John: (Sorry for asking a belated question) It came to me after I've noticed David's answer. So, when do we calculate relativistic velocity. Here, we've just calculated the differences. –  Waffle's Crazy Peanut Sep 22 '12 at 13:40
The number $\gamma$ appears in lots of SR formulae, and it's defined as $\gamma = (1 - v^2/c^2)^{-0.5}$. As long as $\gamma$ is close to one you can approximate the motion as non-relativistic. –  John Rennie Sep 22 '12 at 14:12
So, David expressed the Lorrentz factor as the question's author was eager traveling at $c$..? Thank you John..! (And sorry for those twice comments. They were due to my DEAD dial-up..! –  Waffle's Crazy Peanut Sep 22 '12 at 14:16