Chapter 37, Problem 65P

Another approach to velocity transformations. In Fig. 37-31, reference frames B and C move past reference frame A in the common direction of their x axes. Represent the x components of the velocities of one frame relative to another with a two-letter subscript. For example, v_AB is the x component of the velocity of A relative to B. Similarly, represent the corresponding speed parameters with two-letter subscripts. For example, ß_AB (= v_AB/c) is the speed parameter corresponding to v_AB (a) Show that

${\beta }_{AC}=\frac{{\beta }_{AB}+{\beta }_{BC}}{1+{\beta }_{AB}{\beta }_{BC}}.$

Let M_AB represent the ratio (1 − ß_AB)/(1 + ß_AB), and let M_BC and M_AC represent similar ratios. (b) Show that the relation

M_{AC =} M_ABM_BC

is true by deriving the equation of part (a) from it.

Figure 37-31 Problem 65, 66 and 67.