Chapter 5.9, Problem 23P

To determine

To sow that average speed for the whole trip is the harmonic mean of $uandv$

Answer to Problem 23P

The average speed for the whole trip is the harmonic mean of $uandv$is $\frac{2uv}{u+v}$

Explanation of Solution

Let be the distance be $x$

Given that the half distance be $\frac{x}{2}\text{has}u\text{speed}\text{.}$

Formula:

  $\text{time=}\frac{\text{distance}}{\text{speed}}$

Time taken by speed u

  $t_1=\frac{x/2}{u}$

And remaining half distance be $\frac{x}{2}\text{has}v\text{speed}\text{.}$

Time taken by speed v

  $t_2=\frac{x/2}{v}$

Now, total trip time be $t=t_1+t_2$

  $\begin{array}{c}t=t_1+t_2\\ =\frac{x/2}{u}+\frac{x/2}{v}\\ =\frac{x}{2u}+\frac{x}{2v}\\ =\frac{2v(x)+x(2u)}{4uv}\\ =\frac{2vx+2ux}{4uv}\\ =\frac{x(v+u)}{2uv}\end{array}$

Now, the average

  $\text{Average}\text{}\text{speed=}\frac{\text{total}\text{distance}}{\text{total}\text{time}}$

  $S_{av}=\frac{x}{x\left(\frac{u+v}{2uv}\right)}=\frac{2uv}{u+v}$

Hence, the average speed for the whole trip is the harmonic mean of $uandv$is $\frac{2uv}{u+v}$