When was it possible to observe the special-relativistic modification of Energy?

Question

I am preparing a lecture for high-school students, and was wondering when it would have been possible to observe the special-relativistic modification of the Energy-velocity relation.

In more detail, in classical mechanics the kinetic energy is related to velocity by

$$E^{class}_{kin}=\frac{mv^2}{2}$$ This was first written down in that form by Émilie du Châtelet in the 1720s-1730s. In special relativity, the kinetic energy is given by

$$E^{SRT}_{kin}=\frac{mc^2}{\sqrt{1-\frac{v^2}{c^2}}}-mc^2.$$

We can now Taylor-expand $E^{SRT}_{kin}$ and get $$E^{SRT}_{kin}=\frac{mv^2}{2}+\frac{3 m v^4}{8 c^2} + O(v^6)$$

Now, one could have started from the classical energy formula at the time of Émilie du Châtelet and make a wild guess:

$$E^{guess}_{kin}=\frac{mv^2}{2} + \alpha v^3 + \beta v^4 + O(v^5)$$

Of course, nowadays we know that $\alpha$=0 and $\beta=\frac{3 m}{8 c^2}$ -- but in the early 1700s this was of course unknown. So a natural question is:

My question is: What would have been the earliest possible experiments that could have observed a non-zero value of $\beta$? Could Faraday have seen this with the techniques at his disposal? Or even earlier?

Answer 1

I think that the point where technology reached the relativistic realm was when the first cyclotrons started to accelerate particles to relativistic velocities.

The very first cyclotron was a tabletop device. If memory serves me, within years of the first one a cyclotron was built with a chamber two meters in diamter. Each larger design accelerated to higher velocities.

The operating principle of a cyclotron is quite neat:
The centripetal force that is required to make the particle(s) move in circumnavigating motion is provided by the Lorentz force. Electromagnets are set up in such a way that there is a uniform magnetic field.

In the chamber the circumnavigating particles are spiraling outward; as the particles are being accelerated (twice every circumnavigation), they are going round in wider and wider circles. Upon every acceleration the velocity of the particles increases. However, the angular velcity of the particles remains the same because the particles are going round in wider and wider circles.

As we know: the larger the radius of circumnavigating motion, the larger the required centripetal force. The Lorentz force accommodates that perfectly; the Lorentz force is proportional to the velocity.

So: in the non-relativistic realm the proportionalities are linear, allowing a wonderfully simple cyclotron design.

The designers/builders of cyclotons knew in advance this linear behavior is limited to the non-relativistic realm.

Pushing for relativistic velocities the designers knew they had to take into account that a higher magnetic field density is required when the particles are accelerated up to relativistic velocities.

One approach is to create a specific non-uniform magnetic field. For instance, if you make the magnetic field stronger the farther away from the center then the machine can still have all particles go round at the same angular velocity.

In a ring accelerator, such as the Large Hadron Collider, the field strength of the deflecting magnets is increased over time, in order to keep providing the required deflection. In the case of the Large Hadron Collider: the particles that are injected are already close to the speed of light. Over a period of 20 minutes or so the kinetic energy of the particles is raised further. The strength of the deflecting magnetic fields is adjusted upward continously in order to keep providing the required deflection.

Summerizing:
particle accelerators were the first machines for which it was necessary to take relativistic effect into account. Not taking relativistic effect into account would result in poor performance.

Further reading: 2012 question titled Relativistic centripetal force

Answer 2

I have plotted the function $f(x)=\frac{x^{4}}{c^{2}}$ in Desmos

As you see the graph looks like a pretty straight line until we reach a velocity of 80000m/s.So the poeple back then would have been able to notice the deviation from the $\frac{1}{2}mu^{2}$ only if a object of 1/3 kg was travelling at a velocity of 80000m/s.I dont think even their cannons were powerful enough back then to do such a thing.