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)

$\frac{2v_{1}v_{2}}{v_{1}+v_{2}}$

Answer 2

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 $=\frac{\text{Total distance travelled}}{\text{Total time taken}}$ $=\frac{L}{\frac{L}{2v_{1}}+\frac{L}{2v_{2}}}=\frac{1}{\frac{1}{2}\left[\frac{1}{v_{1}}+\frac{1}{v_{2}}\right]}$ $=\frac{2v_{1}v_{2}}{v_{2}+v_{1}}$