Experimenters in reference frames S and $\text{S}\prime$ measure different values for position and velocity. What about the force on and the acceleration of the particle in FIGURE 36.5? The strength of a force can be measured with a spring scale. The experimenters in reference frames S and $\text{S}\prime$ both see the same reading on the scale (assume the scale has a bright digital display easily seen by all experimenters), so both conclude that the force is the same. That is, $F\prime =F$.

FIGURE 36.5 Experimenters in both reference frames test Newton’s second law by measuring the force on a particle and its acceleration.

We can compare the accelerations measured in the two reference frames by taking the time derivative of the velocity transformation equation $u\prime =u-v$. (We’ll assume, for simplicity, that the velocities and accelerations are all in the x-direction.) The relative velocity v between the two reference frames is constant, with $dv/dt=0$, thus

Experimenters in reference frames S and $\text{S}\prime$ measure different values for an object’s position and velocity, but they agree on its acceleration.

If $F=ma$ in reference frame S, then $F\prime =ma\prime$ in reference frame $\text{S}\prime$. Stated another way, if Newton’s second law is valid in one inertial reference frame, then it is valid in all inertial reference frames. Because other laws of mechanics, such as the conservation laws, follow from Newton’s laws of motion, we can state this conclusion as the Galilean principle of relativity:

The Galilean principle of relativity is easy to state, but to understand it we must understand what is and is not “the same.” To take a specific example, consider the law of conservation of momentum. FIGURE 36.6a shows two particles about to collide. Their total momentum in frame S, where particle 2 is at rest, is $P_{\text{i}}=9.0\text{k}\text{g}\text{m}/\text{s}$. This is an isolated system, hence the law of conservation of momentum tells us that the momentum after the collision will be $P_{\text{f}}=9.0\text{k}\text{g}\text{m}/\text{s}$.

FIGURE 36.6 Total momentum measured in two reference frames.

Full Alternative Text

FIGURE 36.6b has used the velocity transformation to look at the same particles in frame $\text{S}\prime$ in which particle 1 is at rest. The initial momentum in $\text{S}\prime$ is $P_{\text{i}}^{\prime }=-18\text{kg}\text{m}/\text{s}$. Thus it is not the value of the momentum that is the same in all inertial reference frames. Instead, the Galilean principle of relativity tells us that the law of momentum conservation is the same in all inertial reference frames. If $P_{\text{f}}=P_{\text{i}}$ in frame S, then it must be true that $P_{\text{f}}^{\prime }=P_{\text{i}}^{\prime }$ in frame $\text{S}\prime$. Consequently, we can conclude that $P_{\text{f}}^{\prime }$ will be $-18\text{kg}\text{m}/\text{s}$ after the collision in $\text{S}\prime$.