question_answer 32)

                  A vehicle travels half the distance L with speed \[{{V}_{1}}\] and the other half with speed \[{{V}_{2}}\] then its average speed is                 (a) \[\frac{{{V}_{1\,}}\,+\,{{V}_{2}}}{2}\]                       (b) \[\frac{2{{V}_{1\,}}\,+\,{{V}_{2}}}{{{V}_{1\,}}\,+\,{{V}_{2}}}\]                 (c) \[\frac{2{{V}_{1\,}}{{V}_{2}}}{{{V}_{1\,}}\,+\,{{V}_{2}}}\]                  (d) \[\frac{L({{V}_{1\,}}\,+\,{{V}_{2}})}{{{V}_{1\,}}{{V}_{2}}}\]                

Answer:

   (c)  \[{{t}_{1}}\,=\,\frac{L/2}{{{V}_{1}}}\,=\,\frac{L}{2{{V}_{1\,}}}\] and \[{{t}_{2\,}}\,=\,\,\frac{L}{2{{V}_{2}}}\]                 \[\therefore \] \[{{U}_{av}}=\,\frac{L}{{{t}_{1\,}}\,+\,{{t}_{1}}}\,=\,\frac{L}{\frac{L}{2}\left( \frac{1}{{{V}_{1}}}\,+\frac{1}{{{V}_{2}}} \right)}\]                 \[=\frac{2{{V}_{1\,}}{{V}_{2}}}{{{V}_{1\,}}\,+\,{{V}_{2}}}\]