# The 4200 km trip from new york to san francisco takes 7 h flying against the wind, but only 6 h returning. How do you find the speed of the plane in still air and the wind speed?

Apr 26, 2017

Plane speed $= 514 \frac{2}{7} \text{ Km/h" } \approx 514.286$ to 3 decimal places
Wind speed $= 85 \frac{5}{7} \text{ Km/h } \approx 85.714$ to 3 decimal places

#### Explanation:

To solve these equations we need to manipulate so that we have just 1 unknown and its relationship to some values.

Assumption: the speed (velocity) of the wind is constant

Let the speed of the plane be $p$
Let the speed of the wind be $w$
Let time going be in hours be ${t}_{g} \to 7 h$
Let the time returning in hours be ${t}_{r} \to 6 h$

Note that speed = $\left(\text{distance")/("time}\right)$

So $\text{distance "="time "xx" speed}$

Consider going: relative to the ground, the actual speed is $p - w$
Consider returning: relative to the ground the actual speed is $p + w$
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$\textcolor{b l u e}{\text{Determine the ratio of wind speed to plane speed}}$
We need this to change things so that, by substitution, we end up with just 1 unknown.

So for going we have:
$\text{ "4200=7xx(p-w)" } \ldots \ldots \ldots E q u a t i o n \left(1\right)$

And for returning we have:
$\text{ "4200=6xx(p+w)" } \ldots \ldots \ldots \ldots E q u a t i o n \left(2\right)$

Equation both to each other through distance

$7 \times \left(p - w\right) = 4200 = 6 \times \left(p + w\right)$

$7 \times \left(p - w\right) = 6 \times \left(p + w\right)$

$7 p - 7 w = 6 p + 6 w$

Subtract $6 p$ from both sides

$7 p - 6 p = 0 + 6 w$

$p = 6 w \text{ } \ldots \ldots \ldots . . E q u a t i o n \left(3\right)$
$w = \frac{p}{6} \text{ } \ldots \ldots \ldots . . E q u a t i o n \left(4\right)$
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$\textcolor{b l u e}{\text{Determining the actual speeds}}$

Using Equation(3) substitute for $p$ in Equation(1) giving

$4200 = 7 \left(p + w\right) \text{ "->" } 4200 = 7 \left(6 w + w\right)$
$\text{ } 4200 = 49 w$

$w = \frac{4200}{49} = 85.71428 \ldots . = 85 \frac{5}{7} \text{ Km/h}$

Using Equation(4) substitute for $w$ in Equation(1) giving

$4200 = 7 \left(p + w\right) \text{ "->" } 4200 = 7 \left(p + \frac{p}{6}\right)$
$\text{ } 4200 = \frac{49}{6} p$

$p = \frac{6 \times 4200}{49} = 514.2857 \ldots = 514 \frac{2}{7} \text{ Km/h}$