31.1.1 Reference Frames

We introduced reference frames and relative motion in Chapter 4. To remind you, FIGURE 31.2 shows two reference frames labeled A and B. You can think of these as the reference frames in which Alec and Brittney, respectively, are at rest. Frame B moves with velocity v⃗ BA with respect to frame A. That is, an observer (Alec) at rest in A sees the origin of B (Brittney) go past with velocity v⃗ BA. Of course, Brittney would say that Alec has velocity v⃗AB=−v⃗ BA relative to her reference frame. We will stipulate that both reference frames are inertial reference frames, so v⃗ BA is constant.

FIGURE 31.2 Reference frames A and B.

The figure illustrates two reference frames and a particle placed on an x y coordinate plane.
FIGURE 31.2 Full Alternative Text

Figure 31.2 also shows a particle C. Experimenters in frame A measure the motion of the particle and find that its velocity relative to frame A is v⃗CA. At the same instant, experimenters in B find that the particle’s velocity relative to frame B is v ⃗CB. In Chapter 4, we found that v⃗CA and v⃗CB are related by

v⃗CA=v⃗CB+v⃗ BA(31.1)

Equation 31.1, the Galilean transformation of velocity, tells us that the velocity of the particle relative to reference frame A is its velocity relative to frame B plus (vector addition!) the velocity of frame B relative to frame A.

Suppose the particle in Figure 31.2 is accelerating. How does its acceleration a⃗CA, as measured in frame A, compare to the acceleration a⃗CB measured in frame B? We can answer this question by taking the time derivative of Equation 31.1:

d v⃗CAdt=d v⃗CBdt+d v⃗ BAdt

The derivatives of v⃗CA and v⃗CB are the particle’s accelerations a⃗CA and a⃗CB in frames A and B, respectively. But v⃗ BA is a constant velocity, so d v⃗ BA/dt=0⃗. Thus the Galilean transformation of acceleration is simply

a⃗CA=a⃗CB(31.2)

Brittney and Alec may measure different positions and velocities for a particle, but they agree on its acceleration. And if they agree on its acceleration, they must, by using Newton’s second law, agree on the force acting on the particle. That is, experimenters in all inertial reference frames agree about the force acting on a particle.