question_answer

The displacement-time graphs of two bodies  and B are shown in figure. The ratio \[\left( \frac{{{v}_{A}}}{{{v}_{B}}} \right)n\] of velocity of A to velocity of B, is

A)  \[\frac{1}{\sqrt{3}}\]                                    

B)  \[\sqrt{3}\]

C)  \[\frac{1}{3}\]                                  

D)  \[3\]

Correct Answer: C

Solution :

                 Velocity is defined as rate of change of displacement (s). \[\therefore \]  \[v=\frac{\Delta \,s}{\Delta \,t}\] In the given graph,                 Rate of change of displacement                                 = slope of s-t graph                                 \[=\tan \,\theta \] \[\therefore \]  \[\tan {{60}^{o}}=\frac{BD}{OD}={{v}_{B}}\]                 \[\tan {{30}^{o}}=\frac{AC}{OC}={{v}_{A}}\] \[\therefore \]  \[\frac{\tan {{60}^{o}}}{\tan \,{{30}^{o}}}=\frac{{{v}_{B}}}{{{v}_{A}}}\] \[\Rightarrow \]               \[\frac{{{v}_{B}}}{{{v}_{A}}}=\frac{\sqrt{3}}{1/\sqrt{3}}=\frac{3}{1}\] \[\therefore \]  \[{{v}_{A}}:{{v}_{B}}=1:3\]