question_answer

A person completes half of its journey with speed vi and rest half with speed \[{{v}_{2}}\]. The average speed of the person is:

A) \[v=\frac{1}{2}({{v}_{1}}+{{v}_{2}})\]     

B) \[v=\frac{2{{v}_{1}}{{v}_{2}}}{{{v}_{1}}+{{v}_{2}}}\]

C)  \[v=\frac{{{v}_{1}}{{v}_{2}}}{{{v}_{1}}+{{v}_{2}}}\]                         

D)  \[v=\sqrt{{{v}_{1}}{{v}_{2}}}\]

Correct Answer: B

Solution :

Average speed\[=\frac{\text{total}\,\,\text{distance}\,\,\text{travelled}}{\text{total}\,\,\text{time}\,\,\text{taken}}\] Time taken for\[OA=\frac{x}{{{v}_{1}}}\] Time taken for\[AB=\frac{x}{{{v}_{2}}}\] Total time\[=\frac{x}{{{v}_{1}}}+\frac{x}{{{v}_{2}}}\] \[\therefore \]Average speed\[=\frac{2x}{\frac{x}{{{v}_{1}}}+\frac{x}{{{v}_{2}}}}=\frac{2{{v}_{1}}{{v}_{2}}}{{{v}_{1}}+{{v}_{2}}}\]