Relative velocity

The relative velocity v → B ∣ A {displaystyle {vec {v}}_{Bmid A}} (also v → B A {displaystyle {vec {v}}_{BA}} or v → B rel ⁡ A {displaystyle {vec {v}}_{Boperatorname {rel} A}} ) is the velocity of an object or observer B in the rest frame of another object or observer A. The relative velocity v → B ∣ A {displaystyle {vec {v}}_{Bmid A}} (also v → B A {displaystyle {vec {v}}_{BA}} or v → B rel ⁡ A {displaystyle {vec {v}}_{Boperatorname {rel} A}} ) is the velocity of an object or observer B in the rest frame of another object or observer A. We begin with relative motion in the classical, (or non-relativistic, or the Newtonian approximation) that all speeds are much less than the speed of light. This limit is associated with the Galilean transformation. The figure shows a man on top of a train, at the back edge. At 1:00 pm he begins to walk forward at a walking speed of 10 km/h (kilometers per hour). The train is moving at 40 km/h. The figure depicts the man and train at two different times: first, when the journey began, and also one hour later at 2:00 pm. The figure suggests that the man is 50 km from the starting point after having traveled (by walking and by train) for one hour. This, by definition, is 50 km/h, which suggests that the prescription for calculating relative velocity in this fashion is to add the two velocities. The figure displays clocks and rulers to remind the reader that while the logic behind this calculation seem flawless, it makes false assumptions about how clocks and rulers behave. (See The train-and-platform thought experiment.) To recognize that this classical model of relative motion violates special relativity, we generalize the example into an equation: