# On defining mass scales using collisions

In A.P. French's Newtonian Mechanics (pages $$319$$, $$320$$), the author described a way to define a scale for masses. He said,

One can set up an inertial mass scale for a number of objects $$1, 2, 3,…$$ by finding the velocity changes pairwise in such interaction processes and defining the inertial mass ratio, let us say of objects 1 and 2, by $$\frac{m_{2}}{m_{1}}=\frac{|\Delta \textbf{v}|_{1}}{|\Delta \textbf{v}|_{2}}$$ Similarly for objects 1 and 3, $$\frac{m_{3}}{m_{1}}=\frac{|\Delta \textbf{v}'|_{1}}{|\Delta \textbf{v}'|_{3}}$$ and so on. If $$m_{1}$$ is the standard kilogram, we then have an operation to determine the inertial masses of any other objects.

The $$\Delta \textbf{v}$$'s above are the post-collision minus pre-collision velocity, the subscript next to the module indicates the object in question ($$1, 2, 3,...$$).

He went on,

We must, however, do more. If we let objects $$2$$ and $$3$$ interact, then the ratio $$\frac{m_{3}}{m_{2}}=\frac{|\Delta \textbf{v}''|_{2}}{|\Delta \textbf{v}''|_{3}}$$ must be consistent with the same ration obtained from the first two measurements. In fact it is, and this internal experimental consistency then allows us to use the values $$m_{2}, m_{3},...$$ as measures of the inertial masses of the respective objects.

Question: Had there been an inconsistency with the experiment relating object $$2$$ and object $$3$$, i.e. $$\frac{m_{3}}{m_{2}}\ne \frac{|\Delta \textbf{v}''|_{2}}{|\Delta \textbf{v}''|_{3}}$$, would redefining our scale resolve the inconsistency?