The displacement-time graph for two particles A and B are straight lines inclined at angles of 30° and 60° with the time axis. The ratio of velocities of vA : vB is

The correct option is 1:3.

Step-by-Step Explanation:

1. Understanding the physical meaning of the slope:
On a displacement-time graph, the displacement is plotted on the vertical axis (y-axis) and the time is plotted on the horizontal axis (x-axis). The slope of the line representing a particle's motion corresponds to the rate of change of displacement with respect to time, which is defined as velocity ($v$).

2. Relating the slope to the angle of inclination:
Mathematically, the slope of a straight line is equal to the tangent of the angle ($\theta$) that the line makes with the positive direction of the time axis:
$v=\text{Slope}=tan\theta$

3. Calculating individual velocities:
For particle A, the angle of inclination is ${\theta }_A=30°$. Thus, its velocity is:
$v_A=tan30°=\frac{1}{\sqrt{3}}$
For particle B, the angle of inclination is ${\theta }_B=60°$. Thus, its velocity is:
$v_B=tan60°=\sqrt{3}$

4. Finding the ratio of the velocities:
We can now find the ratio of the velocity of particle A to the velocity of particle B:
$\frac{v_A}{v_B}=\frac{tan30°}{tan60°}$
Substituting the trigonometric values:
$\frac{v_A}{v_B}=\frac{1/\sqrt{3}}{\sqrt{3}}=\frac{1}{\sqrt{3}\times \sqrt{3}}=\frac{1}{3}$

Therefore, the ratio of the velocities is 1:3.