If we go slowly, you will never catch up.
K is free parameter in test theories of space-time that span Galilean Relativity, Newtonian Absolute Space and Time and Special Relativity.
In Galileo's/Newton's description of physics, K=0. In Special Relativity, $$K = \frac{1}{c^2}$$. Galileo's and Newton's experiments were too slow and imprecise to distinguish the difference between these quantities.
In the test theory, we simply write K and let experiment decide what value of K best matches the data.
(Test theory) Space-time transformation in one-dimension of space: $$ \Delta x' = \frac{ \Delta x - u \Delta t }{\sqrt{ 1 - K u^2 }} \\ \Delta t' = \frac{ \Delta t - K u \Delta x }{\sqrt{ 1 - K u^2 }}$$
The evidence for a particular value of K comes from precision experiments designed to test aspects of this relation:
(Test theory) Law of composition of velocities in one dimension of space: $$v_3 = \frac{v_1 + v_2}{1 + K v_1 v_2}$$
In 1859, it was empirically discovered that when $$v_1 \approx 10^8 \, \textrm{m} \cdot \textrm{s}^{-1}$$ that $$K= \frac{1}{c^2}$$ was preferred. This formula is derived from the Space-time transformation, so evidence for the latter is evidence for the former.
(Test theory) Law of Proper elapsed time (time dilation): $$\Delta \tau = \sqrt{1 - K v^2} \Delta t$$
Experiments on unstable particles and clock in fast-moving vehicles such as satellites and jets have strongly supported $$K= \frac{1}{c^2}$$. This formula is derived from the Space-time transformation, so evidence for the latter is evidence for the former.
(Test theory) Law of co-moving distances (length contraction): $$L' = \sqrt{1 - K v^2} L_0$$
Observations on unstable particles created at the top of Earth's atmosphere reaching the ground and the engineering of synchroton radiation devices have strongly supported that fast-moving particles treat lengths as contracted and so $$K= \frac{1}{c^2}$$ wins. This formula is derived from the Space-time transformation, as in my earlier post, so evidence for the latter is evidence for the former.
(Test theory) relation between kinetic energy and momentum: $$\frac{p^2}{E - E_0} - 2m = \sqrt{m^2 + Kp^2} -m \approx K \left( \frac{p^2}{2m} - \frac{K p^4}{8 m^3} + \frac{K^2 p^6}{16 m^6} - \dots \right)$$
In the 60's they raced electrons to measure their momentum and energy. $$K= \frac{1}{c^2}$$ wins. This formula is derived from the Space-time transformation, as in my earlier post, so evidence for the latter is evidence for the former.
So we, unlike Newton's assumptions, appear to live in a world where $$K$$ is non-zero and is approximately $$ 1.11265 \times 10^{-17} \, \textrm{m}^{-2} \cdot \textrm{s}^2 = 11.1265 \, \textrm{TJ}^{-1} \cdot \textrm{mg}$$ in SI units. But those numeric value are not natural inventions but represent our human choice to describe the world in certain units.
Because $$K$$ is ubiquitous, it is common for college textbooks to choose a particular unit of time or distance as a standard unit and then choose the other so that when K is expressed in those units is the dimensionless value 1.
Example: In atomic physics, we might choose 1 Bohr radius as the standard unit of length and then $$\sqrt{K} \times 1 \, \textrm{Bohr radius}$$ would be the standard unit of time. In such units, a speed below 1 is slower than light.
Another constant, say the reduced Planck constant, $$\hbar$$ allows expression of mass in terms of the standard unit of length and physical constants. Likewise, adding a third, say Newton's constant of Universal Gravitation, G, means all combinations of length, time and mass can be expressed in combinations of physical constants which presumably results in human size numbers while studying realms of physical behavior where the units are decidedly not human-scale.
The concept is common in university-level instruction in physics for physics majors.
