Question

Mark the correct statements:-

(a) Equations of special relativity are not applicable for small speeds.
(b) Equations of special relativity are applicable for all speeds.
(c) Nonrelativistic equations give exact result for small speeds.
(d) Nonrelativistic equations never give exact result.

Answer 1

(b) Equations of special relativity are applicable for all speeds.
(d) Nonrelativistic equations never give exact result.

 

According to special relativity, if a particle is moving at a very high speed v, its mass

\[m = \gamma m_o,\text{ length }l = \frac{l_o}{\gamma}, \text{  change in time }\Delta t = \gamma\Delta t_o \]

where \[\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\text{ if }v <  < c \Rightarrow \gamma \cong 1\]

that is at non-relativistic speed (small speed),

\[m \cong  m_{o}, l \cong  l_o, \Delta t \cong \Delta t_o\] where \[m_o , l_o\text{ and } \Delta t_o\] are the rest mass, length and time interval respectively. Therefore, relativistic equations are applicable for all speeds. But

\[\gamma =  \left( 1 - \frac{v^2}{c^2} \right)^\frac{- 1}{2} \]

\[ \Rightarrow \gamma = 1 + \frac{v^2}{2 c^2} +  .  .  .  \text{(expanding  binomially)}\]

\[\frac{v^2}{2 c^2} +  .  .  .  = k <  < 1 \text{ if } v <  < c \text{ but still } k > 0\]

Hence, non relativistic equations in which `gamma` factor is taken to be exactly 1 never give exact results.