To solve this problem, we need to understand that the velocity of a particle is given by the slope of its displacement-time graph.$\\$The slope of the graph is $\tan(\theta),$ where $\theta$ is the angle the graph makes with the x-axis.$\\$For the first particle:$\\ \theta_1 = 30^\circ \\ v_1 = \tan(30^\circ) = \frac{1}{\sqrt{3}} \\$For the second particle:$\\ \theta_2 = 45^\circ \\ v_2 = \tan(45^\circ) = 1 \\$The ratio of their velocities is:$\\ \frac{v_1}{v_2} = \frac{\frac{1}{\sqrt{3}}}{1} = \frac{1}{\sqrt{3}} \\$Therefore, the ratio of their respective velocities is $1 : \sqrt{3}. \\$This corresponds to Option 4.