Level of accuracy of experimental confirmation of Newton's second law To what level of accuracy has Newton's second law, as in F = ma, been confirmed by experiment? Have the relativistic adjustments to it been confirmed as well, and to what accuracy?
 A: Any answer to this question is going to be a little complicated, for the following reason. In either Galilean relativity or special relativity, there is no preferred frame of reference. Based on nothing more than this, we can't have equations of motion of the form $x'=f(x)$, because then if the equation held in one frame, it would be guaranteed to fail in another frame. So we expect to have equations of motion of the form $x''=f(x',x)$ (unless there are higher derivatives involved, but higher derivative theories tend not to work very well). Since the equations of motion are guaranteed to be of this form, we can always just take $F=ma$ to be a definition of force, and therefore Newton's second law can only be tested in the sense that we can test for the existence of a preferred frame. People certainly have tested for preferred-frame effects or Lorentz invariance violation, and some of these tests are of insanely high precision, but this probably isn't what you have in mind when you ask for high-precision tests of the second law.
There are a couple of alternatives. (1) We can take $a=F/m$ as a physical law that predicts accelerations, but then if this is not to be vacuous, we have to define force. What this amounts to is that we need to pick some physical theory, such as Newtonian gravity, that predicts forces. Therefore the test of the second law is model-dependent, and it is ambiguous whether any non-null result is actually a violation of the second law or a violation of the other features of the model. In order to interpret such a test, we always need some test theory that predicts something different than the second law. (2) We can test whether force really transforms like a vector and adds like a vector, as we implicitly assume when we state the second law.
The best tests that I know of are mostly gravitational. If you're looking for a test where relativistic effects are negligible, then the best example I know of is the observations that have been used to test MOND. When MOND is used for the purposes for which it was designed, which was to explain rotation curves of galaxies without the need for dark matter, the anomalous accelerations produced are on the order of $10^{-10}$ m/s2. If you don't believe in MOND, and you are satisfied with independent estimates of how much dark matter there is, then you can take this as a rough upper limit on the size of accelerations that violate Newton's second law.
MOND also predicts "external field effects" (EFE), which are basically effects in which forces don't add like vectors. EFE would, for example, produce an effect on the motion of the planets in our solar system due to the existence of the galaxy's external gravitational field. Observations limit any such anomalous acceleration of the planets to no more than about $10^{-14}$ m/s2 (Iorio 2009). 
There are a lot of solar system tests that have very high precision. All of these require relativistic corrections. Lunar laser ranging correctly predicts the orbit of the moon to precisions of about 4 mm over 40 years. The test theory here is PPN, and the PPN parameters are constrained to be less than parts per million.
The Pioneer anomaly has now pretty much been explained as a bogus thermal effect, so it can now be interpreted as a null result. Any anomalous acceleration is limited to about $10^{-9}$ m/s2, compared to a predicted gravitational acceleration of on the order of $10^{-5}$ m/s2 (all of this after taking into account relativistic corrections). Flyby anomalies (anomalous motion of spacecraft diving past a planet in highly elliptical orbits) are also probably bogus, and are no more than about $10^{-4}$ m/s2 compared to a prediction of on the order of 10 m/s2 from theory.
At velocities where relativistic effects are strong, there have been accelerator-based tests of dynamics such as Meyer, 1963, which confirmed $E^2=m^2+p^2$ (in units with c=1) to a precision of about $5\times10^{-4}$ at speeds of about 0.99c. The relativistic velocity-momentum relation has been confirmed to about 0.3% by Zrelov, 1958, in an accelerator-based experiment with protons having $v/c=0.8112$. The really high-precision tests of relativistic dynamics are all quantum-mechanical, though, not classical.
A: $F = m \cdot a$ is not newton's second law of motion it is merely a simplification of newton's 2nd law which when stated word for word is

Impressed force is directly proportional to the rate of change of momentum

mathematically you can state that $$F \space \alpha \space \frac{dp}{dt} $$
Substituting the definition of momentum you get
$F \space \alpha  \space \frac{d(mv)}{dt}$
And since mass is a constant value for many everyday systems we encounter we remove mass from the differential to get our equation $F \space \alpha  \space \space m \cdot\frac{dv}{dt}$
and since dv/dt is a(cceleration) you get F = ma. Which here you can see is a simplification of the second law but not the law itself.
