Relativistic effects When are relativistic effects justifiably negligible? (I know that that is true for 'small velocities', but how small is 'small enough'?) 0.1c, 0.01c, etc.? And how does one properly justify that? I reckon by Taylor expansion of the relativistic formula, but still, how do you set the threshold speed? Thanks.
 A: If $v = kc$ with $k$ small then most special relativistic effects such as time dilation and length contraction are proportional to $$\gamma = \frac{ 1}{ \sqrt{1 - k^2}} \approx 1 + \frac{1}{2} k^2 $$ or it's reciprocal. See Wikipedia here for a more complete discussion. If you consider, for example, $\gamma \le 1.0001$ (a $0.01\%$ effect) to be negligible then this is true if $k \le 0.014$ giving $v \le 9,400,000 MPH = 4,200,000\frac{m}{s}$.
GPS satellites not only need to consider the effects of special relativity but also the effects of general relativity to give accurate positioning.
A: It all depends on what you are considering. Suppose you are timing a sports event like a sprint. Surely, relativistic corrections will be irrelevant since the time scales we are considering are much bigger than the relativistic corrections to any time measurement. 
On the other hand, when we are considering GPS, which is based on sending signals to and from faraway satellites, then relativistic corrections even if they are small do have their importance, especially as they are cumulating over time.
