- Equations of special relativity are applicable for all speeds.
- Nonrelativistic equations never give exact result.
Explanation:
According to special relativity, if a particle is moving at a very high speed v, its mass:
$\text{m}=\gamma\text{ m}_0,$length $\text{l}=\frac{\text{l}_0}{\gamma}, $ change in time $\triangle\text{t}=\gamma\ \triangle\text{t}_0$
where, $\gamma=\frac{1}{\sqrt{1-\frac{\text{v}^2}{\text{c}^2}}}\text{i}\text{f}\ \text{v}<<\text{c}$
$\Rightarrow\gamma\cong1$
that is at non relativistic speed (small speed), $\text{m}\cong\text{m}_0,\text{l}\cong\text{l}_0,\triangle\text{t}\cong\triangle\text{t}_0$
where $\text{m}_0,\text{l}_0$ and $\triangle\text{t}_0$ are the rest mass, length and time interval respectively. Therefore, relativistic equations are applicable for all speeds. But
$\gamma=\Big(1-\frac{\text{v}^2}{\text{c}^2}\Big)^{-\frac{1}{2}}$
$\Rightarrow\gamma=1+\frac{\text{v}^2}{\text{2c}^2}+\cdots$ (expanding binomially)
$\frac{\text{v}^2}{2\text{c}^2}+\cdots=\text{K}<<1$ if $\text{v}<<\text{c}$ but silll $\text{K}>0$
Hence. non relativistic equations. in which $\curlyvee$ factor is taken to be exactly 1 never give exact results.