Hurkyl said:
Ok, so that's speed in an inertial frame. What's relative speed?
Suppose that two objects are not moving relative to each other.
The distance between them in
any frame is unchanging.
In this case, the relative velocity v is zero.
Case II: not (v=0)
Suppose that two objects are in relative motion. Thus, the distance between them is changing.
Choose one object to be at rest, at the origin of its own frame of reference, say F1. Since the two objects are in relative motion, the distance between them is changing. Thus, the coordinates of object 2 are changing in F1.
Let the position of object 2 at the current moment in time, in F1 be denoted by (x,y,z).
Define the position vector of object 2 in F1 as follows:
\vec{r} = xi + yj + zk
Let there be an infinite number of clocks in F1, one at every point. Let all the clocks be synchronized.
Let (x1,y1,z1) be the position of object 2 in F1, the moment when all clocks read 1. Let (x2,y2,z2) be the position of object 2 in F1, the moment when all clocks read 2, and so on.
The relative speed of object 1 to object 2 can be defined as follows:
|\vec{V}| = \frac{|\vec{r2} - \vec{r1}|}{2-1}
Or we can define time of travel using arbitrary clock readings. Let object 2 be located at (x1,y1,z1) when all clocks read A, and later, let it be located at position (x2,y2,z2) when all clocks read B. The clocks increase in their readings, tick at the same rate (by hypothesis), hence B>A. The relative speed of object 2 to object one is the magnitude of the objects velocity vector which is:
\vec{V} = \frac{\vec{r2} - \vec{r1}}{B-A}
Kind regards,
StarThrower
Keep in mind, that the above definition is only useful when the relative speed is constant in time.
If the objects are accelerating relative to each other, then the denominator must be vanishingly small.
