Question

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{2v_{1}+v_{2}}{v_{1}+v_{2}} $
• C. $ \frac{2v_{1}v_{2}}{v_{1}+v_{2}} $
• D. $ \frac{L\left(v_{1}+v_{2}\right)}{v_{1}v_{2}} $

Answer 1

Answer: (C)

Time taken to travel first half distance,
$ t_{1}=\frac{L/2}{v_{1}}=\frac{L}{2v_{1}} $
Time taken to travel second half distance, $ t_{2}=\frac{L/2}{v_{2}}=\frac{L}{2v_{2}} $
Total time taken $ =t_{1}+t_{2}=\frac{L}{2v_{1}}+\frac{L}{2v_{2}} $
Average speed $ =\frac{\text{Total distance travelled}}{\text{Total time taken}} $
$ =\frac{L}{\frac{L}{2v_{1}}+\frac{L}{2v_{2}}}=\frac{1}{\frac{1}{2}\left[\frac{1}{v_{1}}+\frac{1}{v_{2}}\right]} $
$ =\frac{2v_{1}v_{2}}{v_{2}+v_{1}} $