At exactly what speed does Newtonian physics break down?

Answer 1

at any speed above zero. It is just that relativistic effects are not noticable by humans until approaching the speed of light.

Newtonian physics is not correct at any level. It is only a good approximation to the truth, but it is not literally true at any level of reality. Quantum mechanics and relativity have replaced it. Newtonian physics is just used because it simplifies calculations and gives results that are usually (but not always) "accurate enough."

Answer 2

There is no definite "break down" speed at which all of a sudden Newtonian mechanics gives way to Relativistic mechanics.

In the strictest sense, at any non-zero speed, Newtonian physics only gives us an approximation. At very low speeds (speeds much less than the speed of light), the Newtonian approximation is very close to the truth and one does not need to complicate matters by using Relativistic physics. Relativistic mechanics is always true for any speed. Nothing stops you from using it at low speeds...but it is generally not necessary.

There is something called the "gamma function" (also known as the Lorentz factor), which might help you though.
gamma = 1 / sqrt (1 - v^2 / c^2)
This gamma function is present in many relativistic expressions to alter the Newtonian method to account for relativistic speeds.
You can see that as the velocity of the object (v) gets closer and closer to c (the speed of light in a vacuum), the expression gets larger and larger and approaches infinity. As v goes to c, gamma goes to infinity. But when v = 0, gamma = 1.
The larger the value of gamma, the larger the discrepancy between Newtonian and Relativistic mechanics.

For example, the formula for time dilation is,
(delta t) = (delta t_0) * gamma
Where (delta t) and (delta t_0) are the changes in time experienced by two different reference frames moving at some speed (v....from the gamma function) relative to each other.
If gamma = 1 (i.e. when the reference frames are not moving at all with respect to each other, v = 0), then (delta t) and (delta t_0) are equal...there is no difference.
But if gamma is some number larger than 1, the change in time is experiences differently.
You will need to decide for yourself just how large gamma needs to be more the difference in times to be significant...there is not absolute boundary where we say that at this speed Newtonian mechanics works and at this speed you must use relativistic mechanics...it just depends on how large of an error one is willing to tolerate.

Just to give you an idea,
At 50% the speed of light, gamma = 1.155
At 90% the speed of light, gamma = 2.294
At 95% the speed of light, gamma = 3.203
At 99% the speed of light, gamma = 7.089

Answer 3

There is not a definite limit. For speeds of the order of the light speed, let us say 1/10 for some experiments with elemental particles (electrons, protons, etc.), the dynamical behavior follows special relativity laws and not newtonian ones. Newtonian laws are then just an approximation.

Answer 4

The speed of light