If the radius of curvature of the path of two particles of same mass are in the ratio $$3 : 4$$, then in order to have constant centripetal force, their velocities will be in the ratio of:

We need to find the ratio of velocities of two particles with the same mass, given that their radii of curvature are in the ratio 3:4 and the centripetal force is constant.

The centripetal force on a particle of mass $$m$$ moving with speed $$v$$ along a circular path of radius $$r$$ is:

$$F = \frac{mv^2}{r}$$

Since both particles have the same mass and the same centripetal force:

$$\frac{mv_1^2}{r_1} = \frac{mv_2^2}{r_2}$$

The mass cancels:

$$\frac{v_1^2}{r_1} = \frac{v_2^2}{r_2}$$

$$\frac{v_1^2}{v_2^2} = \frac{r_1}{r_2}$$

Taking the square root:

$$\frac{v_1}{v_2} = \sqrt{\frac{r_1}{r_2}} = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2}$$

Therefore, $$v_1 : v_2 = \sqrt{3} : 2$$.

The correct answer is Option (1): $$\sqrt{3} : 2$$.