Chapter 9.6, Problem 23P

To determine

To prove: that the average speed of plane with the wind and against the wind is $\frac{\left({\text{p}}^{\text{2}}{\text{-w}}^{\text{2}}\right)}{\text{p}}\text{km}/\text{h}$ .

Answer to Problem 23P

The average speed is $\frac{\left({\text{p}}^{\text{2}}{\text{-w}}^{\text{2}}\right)}{\text{p}}\text{km}/\text{h}$ .

Explanation of Solution

Given:

A plane has speed $\text{p}\text{km}/\text{h}$ and wind has speed of $\text{w}\text{km}/\text{h}$ .

Concept used:

According to the algebraic identities:

  $\left(\text{a+b}\right)\left(\text{a-b}\right){\text{=a}}^{\text{2}}{\text{-b}}^{\text{2}}$ .

Speed and the rate are same meaning here.

  $\text{Distance=rate}\times \text{time}$ .

When the direction of the speed is same then the speed will add up.

When the direction of the two speed is opposite then the speed is deducted.

The rate of the plane with the wind is:

  $\text{p+w}$ .

the rate of the plane against the wind is:

  $\text{p-w}$ .

Calculation:

If $\text{p}$ is the speed in still air and $\text{w}$ is the speed of the wind, then the rate of the plane with the wind is $\text{p+w}$ and against the wind is $\text{p-w}$ . let $\text{d}$ be the distance in each direction, ${\text{t}}_1$ be travel time with the wind and ${\text{t}}_2$ be the travel time against the wind.

Using $\text{Distance=rate×time}$ gives:

  $\text{d=}\left(\text{p+w}\right){\text{t}}_{\text{1}}$ .

  ${\text{t}}_{\text{1}}=\frac{\text{d}}{\left(\text{p+w}\right)}$ ........ $\left(i\right)$

Similarly,

  $\text{d=}\left(\text{p-w}\right){\text{t}}_2$ .

  ${\text{t}}_{\text{1}}=\frac{\text{d}}{\left(\text{p-w}\right)}$ ......... $\left(\text{ii}\right)$

The total distance is $\text{2d}$ and the total time is ${\text{t}}_{\text{1}}{\text{+t}}_{\text{2}}$ so the average speed is: $\frac{\text{2d}}{{\text{t}}_{\text{1}}{\text{+t}}_{\text{2}}}$ .

Substituting in the equations for ${\text{t}}_1$ and ${\text{t}}_2$ then gives:

  $\frac{\text{2d}}{{\text{t}}_{\text{1}}{\text{+t}}_{\text{2}}}\text{=}\frac{\text{2d}}{\frac{\text{d}}{\text{p+w}}\text{+}\frac{\text{d}}{\text{p-w}}}$ .

  $\Rightarrow \frac{\text{2d}}{\frac{\text{d}\left(\text{p-w}\right)+\text{d}\left(\text{p+w}\right)}{\left(\text{p+w}\right)\left(\text{p-w}\right)}}$ .

  $\Rightarrow \frac{\text{2d}\left(\text{p+w}\right)\left(\text{p-w}\right)}{\text{d}\left(\text{p-w}\right)+\text{d}\left(\text{p+w}\right)}$ .

Factor out $\text{d}$ from numerator and denominator:

  $\Rightarrow \frac{\text{2}\left(\text{p+w}\right)\left(\text{p-w}\right)}{\left(\text{p-w}\right)+\left(\text{p+w}\right)}$ .

According to the algebraic identities:

  $\left(\text{a+b}\right)\left(\text{a-b}\right){\text{=a}}^{\text{2}}{\text{-b}}^{\text{2}}$ .

  $\Rightarrow \frac{\left({\text{p}}^{\text{2}}{\text{-w}}^{\text{2}}\right)}{\text{p}}$ .

Hence, the average speed is $\frac{\left({\text{p}}^{\text{2}}{\text{-w}}^{\text{2}}\right)}{\text{p}}\text{km}/\text{h}$ .