The position-time ( x-t ) graph for two cars, A and B, is shown in...

The velocity of a particle is given by the slope of its position-time graph, i.e., $v = \tan \theta$, where $\theta$ is the angle made by the graph with the time axis.

For car A, the angle with the time axis is $\theta_A = 30^{\circ}$.
Velocity of car A, $v_A = \tan 30^{\circ} = \dfrac{1}{\sqrt{3}}$

For car B, the angle with the position axis is $30^{\circ}$. Therefore, the angle with the time axis is $\theta_B = 90^{\circ} - 30^{\circ} = 60^{\circ}$.
Velocity of car B, $v_B = \tan 60^{\circ} = \sqrt{3}$

The ratio of the velocity of car A to the velocity of car B is:
$\dfrac{v_A}{v_B} = \dfrac{1/\sqrt{3}}{\sqrt{3}} = \dfrac{1}{3}$

Thus, the required ratio is $1 : 3$.

Answer: $1 : 3$