What is the border line that constitutes v c

[chris_0101]

Homework Statement

Relativistic questions usually require an expression to solve which in this case will be called gamma (shown below). Now when solving for time dilation for example, we can multiply proper time by gamma to find the dilated time. However, when a calculator is used and the value of velocity is very small, we modify gamma in order to approximate the dilated time, this is known as the binomial approximation (shown below).

My main question is, what value is the limit or border line should I look out for to determine whether to use the approximation or the original formula

Homework Equations

Gamma = 1/sqrt(1-($$\beta$$²)) where $$\beta$$ = v/c

Binomial approximation:
sqrt(1-($$\beta$$²)) = (1-($$\beta$$²))^1/2 $$\approx$$ 1-(1/2)($$\beta$$²))

 

[G01]

There isn't really a definite value at which the approximation breaks down. The binomial approximation just gets continuously worse and worse as v gets bigger and bigger.

That said:
The rule of thumb usually is that the non-relativistic limit ends at 10% of the speed of light. This is when the results of non-relativistic mechanics differ from relativistic mechanics by 1%.