A vehicle travels half the distance with speed $v$ and the remaining distance with speed $2v$. Its average spe

Question

A vehicle travels half the distance with speed $v$ and the remaining distance with speed $2v$. Its average speed is:

Accepted Answer

1.  **Define Average Speed:** Average speed is defined as the ratio of the total distance traveled to the total time taken .
    $$v_{avg} = \frac{	ext{Total Distance}}{	ext{Total Time}}$$
2.  **Set Up Variables:** Let the total distance be $2d$. The vehicle travels the first half distance ($d$) with speed $v_1 = v$ and the second half distance ($d$) with speed $v_2 = 2v$.
3.  **Calculate Time Intervals:**
    *   Time for the first half ($t_1$) = $\frac{	ext{distance}}{	ext{speed}} = \frac{d}{v}$
    *   Time for the second half ($t_2$) = $\frac{d}{2v}$
4.  **Calculate Total Time:**
    $$T = t_1 + t_2 = \frac{d}{v} + \frac{d}{2v} = \frac{2d + d}{2v} = \frac{3d}{2v}$$
5.  **Calculate Average Speed:**
    $$v_{avg} = \frac{2d}{\frac{3d}{2v}} = \frac{2d 	imes 2v}{3d} = \frac{4v}{3}$$
    *Alternative Formula:* For equal distances covered at speeds $v_1$ and $v_2$, the average speed is the harmonic mean: $v_{avg} = \frac{2v_1v_2}{v_1 + v_2} = \frac{2(v)(2v)}{v + 2v} = \frac{4v^2}{3v} = \frac{4v}{3}$.

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