when an aircraft takes off, it accelerates until it reaches its takeoff speed V. in doing so it uses up a distance R of the runway, where R is proportional to the square of the of the takeoff speed. if V is measured in miles per hour and R is measured in feet, then 0.1639 is the constant of proportionality. if an aircraft has a takeoff speed of about 215 miles per hour, how much runway does it need?

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23-02-2018
Mathematics

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when an aircraft takes off, it accelerates until it reaches its takeoff speed V. in doing so it uses up a distance R of the runway, where R is proportional to the square of the of the takeoff speed. if V is measured in miles per hour and R is measured in feet, then 0.1639 is the constant of proportionality. if an aircraft has a takeoff speed of about 215 miles per hour, how much runway does it need?

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LeonardoVivaldi LeonardoVivaldi · Answer 1 · 2018-03-05T18:09:23+01:00

Answer: 36.22 feet

Explanation:

Two quantities are proportional if and only if their ratio is constant. Since V and the square of R are proportional, then the ratio of V to the square of R is constant. In terms of equation:

[tex] \frac{V}{R^2} = k[/tex]

For some constant k.

Since the constant of proportionality is equal to 0.1639, k = 0.1639. Moreover, since the takeoff speed of the aircraft needs to be 215 miles per hour, the runway needed is will be obtained by solving for R in the following equation:

[tex]\frac{V}{R^2} = k \\ \\ V = kR^2 \\ \\ R^2 = \frac{V}{k} \\ \\ R = \sqrt{\frac{V}{k}} \\ \\ R = \sqrt{\frac{215}{0.1639}} \\ \\ \boxed{R = 36.22 \text{ feet}} [/tex]

Hence, a runway of 36.22 feet is needed for the aircraft to have a takeoff speed of 215 miles per hour.