Question

What is the ratio of the relativistically correct expression for momentum to the classical expression? Under what condition does the deviation become significant?

Short answer

The ratio of the relativistically correct expression for momentum $\left(P_r\right)$to the classical expression $\left(P_c\right)$is equal to the Lorentz factor.

$\frac{P_r}{P_c}=\gamma$

$=\frac{1}{\sqrt{1-{\left({\displaystyle \frac{v}{c}}\right)}^2}}$

The deviation for relativistically correct expression for momentum becomes significant for the non-relativistic speed of particles.

Step-by-step solution

Relativistic Momentum

For the particles moving at relativistic speeds, the mass no longer remains constant but increases with an increase in velocity. This varying mass is called dynamic mass which is Lorentz factor times the rest mass of the particle.

Determination of relativistic correct expression for momentum and classical expression for momentum

The relativistically correct expression for momentum is given as:

$P_r=\gamma mv$

Here is the mass of the particle, c is the speed of light, $\gamma$ is the Lorentz factor and its value is $\sqrt{1-{\left(\frac{v}{c}\right)}^2}$and v is the speed of the particle

The classical expression for momentum is given as:

$P_c=mv$

Determination of the ratio of relativistic correct expression and classical expression for momentum

The ratio of relativistic correct expression and classical expression for momentum is given as:

$$\begin{gathered} \frac{P_r}{P_c}=\frac{\gamma mv}{mv} \\ =\gamma \\ =\frac{1}{\sqrt{1-{\left({\displaystyle \frac{v}{c}}\right)}^2}} \end{gathered}$$

Determination of condition for deviation

When speed of particle becomes very small compared to the speed of light $\left(v\ll c\right)$then Lorentz factor becomes unity and relativistically correct expression for momentum changes to classical formula of momentum.