Question

If the displacement-time graph for the two particles A and B are straight lines inclined at angles of 300 and 600 with the time axis, then find the ratio of the velocities of body A to that of body B.

Answer 1

Appropriate Question:

If the displacement-time graph for the two particles A and B are straight lines inclined at angles of 30° and 60° with the time axis, then find the ratio of the velocities of body A to that of body B

Answer:

The ratio of the velocity of body A to that of body B is 1 : 3

Explanation:

Angle of inclination for particle A, $\sf \theta _A$ = 30°

Angle of inclination for particle B, $\sf \theta _B$ = 60°

The velocity of an object is the rate of change of it's displacement.

$\longmapsto \tt v=\dfrac{dx}{dt}$

In the displacement-time graph, the slope is equal to the velocity.

 slope of the graph, m = tanθ

where θ is the angle of inclination

Velocity of particle A = $\tan \theta _A$

  $\sf v_A=\tan 30^{\circ} \\\\ \sf v_A=\dfrac{1}{\sqrt{3}}$

Velocity of particle B = $\tan \theta _B$

 $\sf v_B=\tan 60^{\circ} \\\\ \sf v_B=\sqrt{3}$

The ratio of the velocities of body A to that of body B :

$\implies \sf \dfrac{v_A}{v_B}=\dfrac{\dfrac{1}{\sqrt{3}}}{\sqrt{3}} \\\\ \implies \sf \dfrac{v_A}{v_B}=\dfrac{1}{\sqrt{3}^2} \\\\ \implies \sf \dfrac{v_A}{v_B}=\dfrac{1}{3}$

Therefore, the ratio of the velocity of body A to that of body B is 1 : 3