This equation depicts the velocity of a subject at position point `P` relative to the reference frame labeled `A`, based on knowledge of the velocity of reference frame `B` relative to the reference frame `A` and based on the knowledge of the velocity of the subject at position point P relative to the reference frame labeled `B`.
This simple examination of relative displacement is the basis for a mathematical depiction of relative velocity as well, and is the fundamental basis of relativity theory.
Essentially this equation helps you examine the situation where you have two observers moving in two separate reference frames. It is important to represent the signs of the relative velocities, so in the picture the x-direction and x-velocity vector direction are positive to the right and negative to the left.
In the picture, a subject is located instantaneously at point `P` and is known to be moving in a reference frame `B` with velocity `v_"x(P/B)"`. The origin of the reference frame `A` is the point `O_A`. If we know the velocity of reference frame `B` relative to reference frame `A`, then we can compute the velocity of the subject `P` relative to the reference frame `A`, which is `v_"x(P/A)"`
`v_"x(P/A)" = v_"x(P/B)" +v_"x(B/A)"`
The relative position of the subject's position point `P` from origin `O_A` is given by:
`x_"P/A" = x_"P/B" + x_"B/A"`1
Knowing that the velocity if the derivative of the position with respect to time, we know that:
`dx_"P/A"/"dt" = dx_"P/B"/"dt" + dx_"B/A"/"dt"`
And this expression is the same as:
`v_"x(P/A)" = v_"x(P/B)" +v_"x(B/A)"`2
In today's world of frequent travel on multi-lane interstates, we can easily picture traveling parallel to a train. In our example, both the train and our car are moving in the same direction and our car is moving a little faster than the train. If we denote our reference frame in which we are moving with the car as `A`, we can also denote the reference frame of the train as `B`. So, sitting in the car we see motion in reference frame `A` and people sitting on the train see motion in reference frame `B`. Let's let our car's velocity in reference frame `A` be 75 mph and the train's velocity in reference frame be 65 mph. Then the velocity of the train, reference frame `B`, relative to the car's reference frame `A` is -10 mph. This makes sense as the train is going 10 mph slower in the same direction.
`v_x(B/A) = -10 mph
On the train is a passenger walking forward on the train at a brisk 3 mph, and so the passenger's velocity in the train's reference frame is 3 mph..
`v_x(P/B) = 3 mph
Then we can calculate the velocity of the passenger relative to reference frame `A`, relative to us in the car, the reference frame in which our car is moving, as:
`v_"x(P/A)" = v_"x(P/B)" +v_"x(B/A)" = 3 mph + (-10 mph) = -7 mph`
This simply means that the passenger appears to us as we travel in the car as if they are moving backwards at 7 mph.
Relative Position in One Dimension