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

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.

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:

$P_c=mv$

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

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

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.