Lesson Airspeed, Time, and Distance

Source code of 'Airspeed, Time, and Distance'

This Lesson (Airspeed, Time, and Distance) was created by by oberobic(2304)

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Problem: A light aircraft flies from A to B, 450km away, and returns from B and A in a total time of 5 hours and 30 minutes. Suppose that during the whole journey there is a constant wind blowing from A to B. The speed of the aircraft in still air is 165 km/h. Find the speed of the wind.
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Solution: The basic approach to solving problems such as these is to begin with the fundamental distance equation and plug in the known values. Then solve for the unknowns.
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d = rt is the basic distance equation.
d = 450 km
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We are told the wind is blowing from A to B, so on the trip from A to B the airplane's speed is 165 km/h + w, where 'w' is the wind speed in km/h. That means the speed across the ground (the ground speed) is faster than the indicated airspeed. A wind from behind is called a 'tailwind.'
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Similarly, the ground speed going from B to A will be reduced by the wind. The indicated airspeed will still be 165, but the speed across the ground will be 165 - w. Such a wind is called a 'headwind.'
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We must assume the distance from A to B equals the distance from B to A. That is a reasonable mathematical assumption, although there are many factors in aviation that could force the path to be quite different going and coming. 
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We assume the plane is flying at an indicated airspeed of 165 km/h in both directions.
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We are told the roundtrip time is 5 hr 30 min, which = 5.5 hr.
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A to B trip:
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{{{ 450 = (165+w)*t }}}
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{{{ t = 450/(165+w) }}}
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B to A trip:
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{{{ 450 = (165-w)*(5.5-t) }}}
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{{{ 5.5-t = 450/(165-w) }}}
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{{{ -t = 450/(165-w) - 5.5 }}}
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{{{ t = -450/(165-w) + 5.5 }}}
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{{{ t = t }}}
so
{{{ 450/(165+w) = -450/(165-w) + 5.5 }}}
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{{{ 450/(165+w) + 450/(165-w) = 5.5 }}}
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{{{ ( 450*(165-w) + 450*(165+w) ) / ( (165+w)*(165-w) ) = 5.5 }}}
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{{{ ( 450*(165-w) + 450*(165+w) ) = ( (165+w)*(165-w) ) * 5.5 }}}
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{{{ 450*(165-w+165+w) = (165^2 -165w + 165w -w^2) * 5.5 }}}
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{{{ 450*330 = (165^2 -165w + 165w -w^2) * 5.5 }}}
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{{{ 450*330/5.5 = (165^2 -165w + 165w -w^2) }}}
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{{{ 27000 = -w^2 + 165^2 }}}
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{{{ w^2 = 165^2 -27000 }}}
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{{{ w^2 = 27225 -27000 =225 }}}
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{{{ w = 15 }}}
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Note that sqrt(225) = + or - 15, but a negative wind speed is not applicable.
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Substitute w = 15
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{{{ 450 = (165+w)*t }}}
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{{{ 450 = (165+15)*t }}}
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{{{ 450 = 180t }}}
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{{{ 180t = 450 }}}
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{{{ t = 450/180 = 45/18 = 5/2 = 2.5 }}}
The flying time from A to B with a tailwind is 2.5 hr.
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{{{ 5.5-t = 5.5-2.5 = 3 }}}
The flying time from B to A with a headwind is 3 hr.
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Check the distances traveled to be sure this is the answer.
{{{ (165+15)*2.5 = 450 }}}
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{{{ (165-15)*3 = 450 }}}
Correct.
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Answer: The wind speed is 15 km/h.