FIGURE 4.17 shows Amy and Bill watching Carlos on his bicycle. According to Amy, Carlos’s velocity is $v_x=5\text{m/s}$. Bill sees the bicycle receding in his rearview mirror, in the negative x-direction, getting 10 m farther away from him every second. According to Bill, Carlos’s velocity is $v_x=-10\text{m/s}$. Which is Carlos’s true velocity?

FIGURE 4.17 Velocities in Amy’s reference frame.

Velocity is not a concept that can be true or false. Carlos’s velocity relative to Amy is ${\left(v_x\right)}_{\text{CA}}=5\text{m}\text{/}\text{s}$, where the subscript notation means “C relative to A.” Similarly, Carlos’s velocity relative to Bill is ${\left(v_x\right)}_{\text{CB}}=-10\text{m}\text{/}\text{s}$. These are both valid descriptions of Carlos’s motion.

It’s not hard to see how to combine the velocities for one-dimensional motion:

(4.16)

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We’ll justify this relationship later in this section and then extend it to two-dimensional motion.

Equation 4.16 tells us that the velocity of C relative to B is the velocity of C relative to A plus the velocity of A relative to B. Note that

because if B is moving to the right relative to A, then A is moving to the left relative to B. In Figure 4.17, Bill is moving to the right relative to Amy with ${\left(v_x\right)}_{\text{BA}}=15\text{m}\text{/}\text{s}$, so ${\left(v_x\right)}_{\text{AB}}=-15\text{m}\text{/}\text{s}$. Knowing that Carlos’s velocity relative to Amy is 5 m/s, we find that Carlos’s velocity relative to Bill is, as expected, ${\left(v_x\right)}_{\text{CB}}={\left(v_x\right)}_{\text{CA}}+{\left(v_x\right)}_{\text{AB}}=5\text{m/s}+(-15)\text{m}\text{/}\text{s}=-10\text{m}\text{/}\text{s}$.