The Galilean transformation

We now consider how the values of the position, velocity, and acceleration of a particle differ in two inertial frames that move relative to each other at constant velocity. Figure 3.3 shows two such frames, S and S'. Frame S' moves at constant velocity u relative to frame S. We assume that the origins О and O' coincided at t = 0. The positions of a point P relative to the two frames are related by

FIGURE 3.3

r ' = r − u t (3.4)

This vector equation is equivalent to three equations in terms of the components. An often encountered special case occurs when the x and x' axes coincide and frame S' moves at constant velocity + u i along the x axis of S, as shown in Fig. 3.4. The у and z coordinates of P are the same for both coordinate systems. The figure shows that x' = x − ut. Therefore, for this special case,

x' = x − ut, y'=y, z'=z, t'=t. (3.5)

FIGURE 3.4

Equation 3.5 relates the coordinates of a particle in two inertial frames moving relative to each other at constant velocity. It is called the Galilean transformation of coordinates.

The velocity of the particle may be found by taking the derivative of Eq.3.4 with respect to time:

v ' = v − u (3.6)

The time derivative of Eq.3.6 yields the accelerations (note d u /dt= 0):

a = a' (3.7)

Observers in all inertial frames would assign the same acceleration to the particle. To see the meaning of Eq.3.7, consider a simple experiment. In Fig. 3.5 a, a ball is thrown up vertically from a railcar (frame S'). An observer in this frame will see the ball move only along the y' axis with the acceleration due to gravity. Now suppose the railcar moves at constant velocity +u along the x axis of the ground frame S, as in Fig. 3.5 b. In this frame, the ball always has the fixed horizontal velocity +u. The path described in S is a combination of the fixed horizontal velocity and the vertical accelerated motion. The path is parabolic. Although the horizontal velocity of the ball is different in S and S', its acceleration is the same in both. If someone on the ground were to throw a ball vertically up, an observer in S' would see a parabolic path.

FIGURE 3.5

In both frames, either path, vertical or parabolic, is associated with the same acceleration. This simple experiment would not allow us to claim that one frame is fixed, while the other is moving. In fact no experiments allow us to distinguish between inertial frames. This is the essence of the Galilean principle of relativity which states:

The laws of mechanics have the same form in all inertial reference frames.

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